CALCU-1. Find the zeros of the polynomial function

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Solution by Steps

step 1

To find the zeros of the polynomial function

f(x) = x^4 - 2x^3 + 4x^2 - 8x

, we start by factoring out the common term

x

step 2

Factor out

x

from the polynomial:

x(x^3 - 2x^2 + 4x - 8) = 0

step 3

Set each factor equal to zero:

x = 0

and

x^3 - 2x^2 + 4x - 8 = 0

step 4

Solve the cubic equation

x^3 - 2x^2 + 4x - 8 = 0

. This can be done by finding the roots of the cubic polynomial

step 5

Using the Asksia-LL calculator result, the roots of the cubic polynomial

x^3 - 2x^2 + 4x - 8 = 0

are

x = 2

and

x = \pm 2i

step 6

Combine all the roots:

x = 0

,

x = 2

, and

x = \pm 2i

Answer

The zeros of the polynomial function

f(x) = x^4 - 2x^3 + 4x^2 - 8x

are

x = 0

,

x = 2

, and

x = \pm 2i

.

Key Concept

Finding the zeros of a polynomial function

Explanation

To find the zeros of a polynomial function, factor the polynomial and solve for the values of

x

that make each factor equal to zero.

Solution by Steps

step 1

To simplify

i^{37}

, we start by recognizing that

i

is the imaginary unit, where

i = \sqrt{-1}

step 2

The powers of

i

cycle every 4 terms:

i, -1, -i, 1

step 3

To find

i^{37}

, we determine the remainder of

37

divided by

4

:

37 \mod 4 = 1

step 4

Therefore,

i^{37} = i^1 = i

step 1

To simplify

i^{77}

, we use the same approach as before

step 2

The powers of

i

cycle every 4 terms:

i, -1, -i, 1

step 3

To find

i^{77}

, we determine the remainder of

77

divided by

4

:

77 \mod 4 = 1

step 4

Therefore,

i^{77} = i^1 = i

Answer

i^{37} = i

and

i^{77} = i

Key Concept

Powers of the imaginary unit

i

Explanation

The powers of

i

cycle every 4 terms:

i, -1, -i, 1

. By finding the remainder of the exponent divided by 4, we can determine the simplified form of any power of

i

.

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Solution by Steps

step 1

To find the remainder of the polynomial division

\left(6 x^{4}+15 x^{3}-5 x^{2}+7 x+3\right) \div (x+3)

, we use polynomial long division

step 2

Divide the leading term of the dividend

6x^4

by the leading term of the divisor

x

, which gives

6x^3

step 3

Multiply the entire divisor

(x+3)

by

6x^3

to get

6x^4 + 18x^3

. Subtract this from the original polynomial:

\left(6 x^{4}+15 x^{3}-5 x^{2}+7 x+3\right) - \left(6 x^{4} + 18 x^{3}\right) = -3x^3 - 5x^2 + 7x + 3

step 4

Repeat the process with the new polynomial

-3x^3 - 5x^2 + 7x + 3

. Divide

-3x^3

by

x

to get

-3x^2

. Multiply the divisor by

-3x^2

to get

-3x^3 - 9x^2

. Subtract:

\left(-3x^3 - 5x^2 + 7x + 3\right) - \left(-3x^3 - 9x^2\right) = 4x^2 + 7x + 3

step 5

Continue with

4x^2 + 7x + 3

. Divide

4x^2

by

x

to get

4x

. Multiply the divisor by

4x

to get

4x^2 + 12x

. Subtract:

\left(4x^2 + 7x + 3\right) - \left(4x^2 + 12x\right) = -5x + 3

step 6

Finally, divide

-5x

by

x

to get

-5

. Multiply the divisor by

-5

to get

-5x - 15

. Subtract:

\left(-5x + 3\right) - \left(-5x - 15\right) = 18

step 7

The remainder of the division is

18

Answer

The remainder is

18

Key Concept

Polynomial Long Division

Explanation

Polynomial long division is a method used to divide polynomials, similar to the long division of numbers. The remainder is the part of the dividend that is left over after the division process.

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Solution by Steps

step 1

To find the rational zeros of the polynomial

x^4 - x^3 + 2x^2 - 8x - 48

, we use the Rational Root Theorem. This theorem states that any rational solution,

\frac{p}{q}

, of the polynomial equation must be a factor of the constant term (-48) divided by a factor of the leading coefficient (1)

step 2

The factors of the constant term (-48) are:

\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48

. The factors of the leading coefficient (1) are:

\pm 1

. Therefore, the possible rational zeros are:

\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 16, \pm 24, \pm 48

step 3

We test these possible rational zeros by substituting them into the polynomial

x^4 - x^3 + 2x^2 - 8x - 48

and checking if the result is zero

step 4

Testing

x = -2

:

(-2)^4 - (-2)^3 + 2(-2)^2 - 8(-2) - 48 = 16 + 8 + 8 + 16 - 48 = 0

. Therefore,

x = -2

is a rational zero

step 5

Testing

x = 3

:

(3)^4 - (3)^3 + 2(3)^2 - 8(3) - 48 = 81 - 27 + 18 - 24 - 48 = 0

. Therefore,

x = 3

is a rational zero

step 6

We have found two rational zeros:

x = -2

and

x = 3

Answer

The rational zeros of the polynomial

x^4 - x^3 + 2x^2 - 8x - 48

are

x = -2

and

x = 3

.

Key Concept

Rational Root Theorem

Explanation

The Rational Root Theorem helps us find possible rational zeros of a polynomial by considering the factors of the constant term and the leading coefficient.

Solution by Steps

step 1

To simplify the fraction

\frac{5+2i}{1+6i}

, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of

1+6i

is

1-6i

step 2

Multiply the numerator and the denominator by

1-6i

:

\frac{(5+2i)(1-6i)}{(1+6i)(1-6i)}

step 3

Expand the numerator:

(5+2i)(1-6i) = 5 - 30i + 2i - 12i^2 = 5 - 28i + 12 = 17 - 28i

(Note:

i^2 = -1

)

step 4

Expand the denominator:

(1+6i)(1-6i) = 1 - 36i^2 = 1 + 36 = 37

(Note:

i^2 = -1

)

step 5

Combine the results:

\frac{17 - 28i}{37} = \frac{17}{37} - \frac{28i}{37}

step 6

Simplify the fraction:

\frac{17}{37} - \frac{28i}{37} \approx 0.459459459459459459459459459459459459459459459459459459459459459 - 0.756756756756756756756756756756756756756756756756756756756756756i

step 7

Express in polar form:

\sqrt{\frac{29}{37}} \left( \cos\left(-\tan^{-1}\left(\frac{28}{17}\right)\right) + i \sin\left(-\tan^{-1}\left(\frac{28}{17}\right)\right) \right)

Answer

\frac{17}{37} - \frac{28i}{37}

or approximately

0.459459459459459459459459459459459459459459459459459459459459459 - 0.756756756756756756756756756756756756756756756756756756756756756i

Key Concept

Simplifying complex fractions

Explanation

To simplify a complex fraction, multiply the numerator and the denominator by the conjugate of the denominator, then simplify the resulting expression.

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Solution by Steps

step 1

The given polynomial is

x^4 + 2x^2 - 24

. We need to factorize it

step 2

First, we look for possible factorizations. We can rewrite the polynomial as

(x^2)^2 + 2(x^2) - 24

step 3

Let

y = x^2

. Then the polynomial becomes

y^2 + 2y - 24

step 4

We factorize

y^2 + 2y - 24

as

(y - 4)(y + 6)

step 5

Substituting back

y = x^2

, we get

(x^2 - 4)(x^2 + 6)

step 6

Further factorizing

x^2 - 4

as

(x - 2)(x + 2)

, we get

(x - 2)(x + 2)(x^2 + 6)

step 7

The complete factorization of

x^4 + 2x^2 - 24

is

(x - 2)(x + 2)(x^2 + 6)

Answer

The factors of

x^4 + 2x^2 - 24

are

(x - 2)(x + 2)(x^2 + 6)

.

Key Concept

Factorization of polynomials

Explanation

To factorize a polynomial, we can use substitution to simplify the expression and then factorize it step by step. In this case, we substituted

y = x^2

to factorize the quadratic polynomial and then substituted back to get the factors in terms of

x

.

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Solution by Steps

step 1

Identify the vertical asymptotes by setting the denominator equal to zero and solving for

x

step 2

The denominator of the function

f(x) = \frac{3x+6}{x^2-9}

is

x^2 - 9

. Set

x^2 - 9 = 0

step 3

Solve for

x

:

x^2 - 9 = 0 \implies x^2 = 9 \implies x = \pm 3

. Therefore, the vertical asymptotes are

x = 3

and

x = -3

step 4

Identify the horizontal asymptote by comparing the degrees of the numerator and the denominator

step 5

The degree of the numerator

3x + 6

is 1, and the degree of the denominator

x^2 - 9

is 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is

y = 0

Answer

The vertical asymptotes are

x = 3

and

x = -3

, and the horizontal asymptote is

y = 0

.

Key Concept

Asymptotes of Rational Functions

Explanation

Vertical asymptotes occur where the denominator is zero and the numerator is non-zero. Horizontal asymptotes are determined by the degrees of the numerator and denominator.

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Solution by Steps

step 1

To determine the domain of the function

f(x) = \log(x + 2) - 4

, we need to find the values of

x

for which the argument of the logarithm is positive

step 2

The argument of the logarithm is

x + 2

. For

\log(x + 2)

to be defined, x + 2 > 0

step 3

Solving the inequality x + 2 > 0 gives x > -2. Therefore, the domain of

f(x)

is

(-2, \infty)

step 4

To determine the range of the function

f(x) = \log(x + 2) - 4

, we note that the logarithmic function

\log(x + 2)

can take any real value

step 5

Since

\log(x + 2)

can be any real number, subtracting 4 from it will also result in any real number. Therefore, the range of

f(x)

is

(-\infty, \infty)

Answer

The domain of

f(x) = \log(x + 2) - 4

is

(-2, \infty)

and the range is

(-\infty, \infty)

.

Key Concept

Domain and Range of Logarithmic Functions

Explanation

The domain of a logarithmic function

\log(x + a)

is determined by the condition x + a > 0. The range of a logarithmic function is all real numbers, as the logarithm can take any real value.

Solution by Steps

step 1

Identify the given values: Present Value (PV) = $25000, Interest Rate (i) = 8\%, Compounding Frequency (f) = 52 (weekly), and Number of Years (n) = 20

step 2

Convert the interest rate to a decimal:

i = \frac{8}{100} = 0.08

step 3

Calculate the total number of compounding periods:

f \cdot n = 52 \cdot 20 = 1040

step 4

Use the future value formula for compound interest:

FV = PV \left(1 + \frac{i}{f}\right)^{f \cdot n}

step 5

Substitute the values into the formula:

FV = 25000 \left(1 + \frac{0.08}{52}\right)^{1040}

step 6

Simplify the expression inside the parentheses:

1 + \frac{0.08}{52} = 1 + 0.00153846 \approx 1.00153846

step 7

Raise the simplified expression to the power of 1040:

1.00153846^{1040} \approx 4.944

step 8

Multiply by the present value to find the future value:

FV = 25000 \cdot 4.944 \approx 123600

Answer

The amount in the account after 20 years is approximately $123,600.

Key Concept

Compound Interest

Explanation

Compound interest is calculated by applying the interest rate to the initial principal, which then accumulates interest over multiple periods. In this case, the interest is compounded weekly, leading to a significant increase in the future value over 20 years.

Solution by Steps

step 1

The given equation is

\log_h i = j

step 2

To express this in exponential form, we use the definition of logarithms:

\log_b a = c

is equivalent to

b^c = a

step 3

Applying this definition, we get

h^j = i

Answer

h^j = i

Key Concept

Exponential Form of a Logarithmic Equation

Explanation

The exponential form of a logarithmic equation

\log_b a = c

is

b^c = a

. This transformation allows us to switch between logarithmic and exponential representations.

Solution by Steps

step 1

We need to evaluate the logarithm of

\frac{1}{125}

with base 5

step 2

Recall that

\log_b(a) = c

means

b^c = a

step 3

We set up the equation:

5^c = \frac{1}{125}

step 4

Notice that

125 = 5^3

, so

\frac{1}{125} = 5^{-3}

step 5

Therefore,

5^c = 5^{-3}

step 6

Since the bases are the same, we can equate the exponents:

c = -3

step 7

Thus,

\log_5\left(\frac{1}{125}\right) = -3

Answer

-3

Key Concept

Logarithm Evaluation

Explanation

To evaluate

\log_5\left(\frac{1}{125}\right)

, we recognize that

\frac{1}{125}

is the same as

5^{-3}

, leading us to the conclusion that the logarithm is

-3

.

Solution by Steps

step 1

We start by expanding the logarithmic expression

\log_{2} \left( \frac{x^{2} y^{4}}{4 m n^{3}} \right)

step 2

Using the properties of logarithms, we can separate the terms:

\log_{2} \left( \frac{x^{2} y^{4}}{4 m n^{3}} \right) = \log_{2} (x^{2} y^{4}) - \log_{2} (4 m n^{3})

step 3

Further expanding, we get:

\log_{2} (x^{2} y^{4}) = \log_{2} (x^{2}) + \log_{2} (y^{4})

and

\log_{2} (4 m n^{3}) = \log_{2} (4) + \log_{2} (m) + \log_{2} (n^{3})

step 4

Applying the power rule of logarithms, we have:

\log_{2} (x^{2}) = 2 \log_{2} (x)

,

\log_{2} (y^{4}) = 4 \log_{2} (y)

, and

\log_{2} (n^{3}) = 3 \log_{2} (n)

step 5

Substituting these into our expression, we get:

2 \log_{2} (x) + 4 \log_{2} (y) - (\log_{2} (4) + \log_{2} (m) + 3 \log_{2} (n))

step 6

Since

\log_{2} (4) = 2

, the final expanded form is:

2 \log_{2} (x) + 4 \log_{2} (y) - 2 - \log_{2} (m) - 3 \log_{2} (n)

Answer

2 \log_{2} (x) + 4 \log_{2} (y) - 2 - \log_{2} (m) - 3 \log_{2} (n)

Key Concept

Logarithmic Expansion

Explanation

The properties of logarithms allow us to expand a complex logarithmic expression into simpler terms by using the product, quotient, and power rules.

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Solution by Steps

Question 13: Evaluate $\log_{3} 7$

step 1

To evaluate

\log_{3} 7

, we use the change of base formula:

\log_{a} b = \frac{\log b}{\log a}

step 2

Applying the change of base formula:

\log_{3} 7 = \frac{\log 7}{\log 3}

step 3

Using a calculator to find the values:

\log 7 \approx 0.8451

and

\log 3 \approx 0.4771

step 4

Dividing the values:

\frac{0.8451}{0.4771} \approx 1.7712

Answer

\log_{3} 7 \approx 1.7712

Key Concept

Change of Base Formula

Explanation

The change of base formula allows us to evaluate logarithms with any base by converting them to a ratio of common logarithms (base 10 or base e).

Question 14: Solve for $x$, $3^{2x+1}=121$

step 1

To solve

3^{2x+1} = 121

for

x

, we first take the natural logarithm of both sides:

\ln(3^{2x+1}) = \ln(121)

step 2

Using the logarithm power rule:

(2x+1) \ln(3) = \ln(121)

step 3

Solving for

x

:

2x+1 = \frac{\ln(121)}{\ln(3)}

step 4

Simplifying

\ln(121) = 2 \ln(11)

:

2x+1 = \frac{2 \ln(11)}{\ln(3)}

step 5

Isolating

x

:

2x = \frac{2 \ln(11)}{\ln(3)} - 1

step 6

Dividing by 2:

x = \frac{\ln(11)}{\ln(3)} - \frac{1}{2}

Answer

x = \frac{\ln(11)}{\ln(3)} - \frac{1}{2}

Key Concept

Logarithmic Equations

Explanation

To solve exponential equations, we can take the logarithm of both sides and use logarithmic properties to isolate the variable.

Generated Graph

Solution by Steps

step 1

We start with the formula for continuous compounding:

A = P e^{rt}

, where

A

is the amount,

P

is the principal,

r

is the rate, and

t

is the time

step 2

Given

P = 18000

,

r = 0.05

, and

A = 3P = 54000

, we set up the equation:

54000 = 18000 e^{0.05t}

step 3

Divide both sides by

18000

:

3 = e^{0.05t}

step 4

Take the natural logarithm of both sides:

\ln(3) = 0.05t

step 5

Solve for

t

:

t = \frac{\ln(3)}{0.05} \approx 21.97

Answer

It takes approximately 21.97 years for the investment to triple.

Key Concept

Continuous compounding formula

Explanation

The continuous compounding formula

A = P e^{rt}

is used to find the time it takes for an investment to grow to a certain amount.

Question 16 16. There are 200 bacteria in Tricia's mouth. The relative growth rate is

5\%

per minute. How many will there be in one day?

step 1

We use the formula for continuous growth:

N = N_0 e^{rt}

, where

N

is the final amount,

N_0

is the initial amount,

r

is the growth rate, and

t

is the time

step 2

Given

N_0 = 200

,

r = 0.05

, and

t = 1440

minutes (1 day), we set up the equation:

N = 200 e^{0.05 \times 1440}

step 3

Calculate the exponent:

0.05 \times 1440 = 72

step 4

Substitute and solve:

N = 200 e^{72} \approx 3.71734 \times 10^{33}

Answer

There will be approximately

3.71734 \times 10^{33}

bacteria in one day.

Key Concept

Continuous growth formula

Explanation

The continuous growth formula

N = N_0 e^{rt}

is used to calculate the number of bacteria after a certain period given a constant growth rate.

Question 17 17. Bowmenium has a half-life of 25 minutes. If you start with a sample of 200 g, how much will be left after 2 hours?

step 1

We use the formula for exponential decay:

N = N_0 e^{kt}

, where

N

is the final amount,

N_0

is the initial amount,

k

is the decay constant, and

t

is the time

step 2

The decay constant

k

is related to the half-life

T_{1/2}

by

k = -\frac{\ln(2)}{T_{1/2}}

. Given

T_{1/2} = 25

minutes,

k = -\frac{\ln(2)}{25} \approx -0.0277

step 3

Given

N_0 = 200

g and

t = 120

minutes (2 hours), we set up the equation:

N = 200 e^{-0.0277 \times 120}

step 4

Calculate the exponent:

-0.0277 \times 120 \approx -3.324

step 5

Substitute and solve:

N = 200 e^{-3.324} \approx 7.17937

g

Answer

Approximately 7.17937 g of Bowmenium will be left after 2 hours.

Key Concept

Exponential decay formula

Explanation

The exponential decay formula

N = N_0 e^{kt}

is used to calculate the remaining amount of a substance after a certain period given its half-life.

Solution by Steps

step 1

Solve the system of equations:

\begin{cases} x^2 + y^2 = 36 \\ x^2 - y^2 = -14 \end{cases}

step 2

Add the two equations to eliminate

y^2

:

x^2 + y^2 + x^2 - y^2 = 36 - 14 \\ 2x^2 = 22 \\ x^2 = 11 \\ x = \pm \sqrt{11}

step 3

Substitute

x = \pm \sqrt{11}

back into the first equation to solve for

y

:

(\sqrt{11})^2 + y^2 = 36 \\ 11 + y^2 = 36 \\ y^2 = 25 \\ y = \pm 5

step 4

The solutions are:

(x, y) = (\sqrt{11}, 5), (\sqrt{11}, -5), (-\sqrt{11}, 5), (-\sqrt{11}, -5)

Answer

The system has four solutions:

(\sqrt{11}, 5)

,

(\sqrt{11}, -5)

,

(- \sqrt{11}, 5)

,

(- \sqrt{11}, -5)

Question 19

step 1

Solve the system of equations:

\begin{cases} 15x + 10y = 20 \\ 6x + 4y = 7 \end{cases}

step 2

Simplify the second equation by dividing by 2:

3x + 2y = 3.5

step 3

Multiply the simplified second equation by 5 to align coefficients with the first equation:

15x + 10y = 17.5

step 4

Compare the two equations:

15x + 10y = 20 \\ 15x + 10y = 17.5

Since

20 \neq 17.5

, the system has no solutions

Answer

The system has no solutions

Key Concept

Solving systems of equations

Explanation

For the first system, we used substitution and elimination to find the solutions. For the second system, we found that the equations are inconsistent, leading to no solutions.

Solution by Steps

step 1

Let

v

be the speed of the boat in still water and

c

be the speed of the current. The effective speed upstream is

v - c

and downstream is

v + c

step 2

The distance each way is 4 miles. The time taken upstream is 3 hours, so we have the equation:

4 = 3(v - c)

step 3

The time taken downstream is 45 minutes, which is 0.75 hours, so we have the equation:

4 = 0.75(v + c)

step 4

Solving the first equation for

v - c

:

v - c = \frac{4}{3}

step 5

Solving the second equation for

v + c

:

v + c = \frac{4}{0.75} = \frac{16}{3}

step 6

Adding the two equations:

(v - c) + (v + c) = \frac{4}{3} + \frac{16}{3} \Rightarrow 2v = \frac{20}{3} \Rightarrow v = \frac{10}{3}

step 7

Subtracting the first equation from the second:

(v + c) - (v - c) = \frac{16}{3} - \frac{4}{3} \Rightarrow 2c = \frac{12}{3} \Rightarrow c = 2

Answer

The speed of the current is 2 miles per hour.

Question 21

step 1

Let

x

be the amount invested in the

3\%

account. The remaining amount,

10000 - x

, is invested in the

11\%

account

step 2

The interest earned from the

3\%

account is

0.03x

step 3

The interest earned from the

11\%

account is

0.11(10000 - x)

step 4

The total interest earned is

500

, so we have the equation:

0.03x + 0.11(10000 - x) = 500

step 5

Expanding and simplifying the equation:

0.03x + 1100 - 0.11x = 500 \Rightarrow -0.08x + 1100 = 500 \Rightarrow -0.08x = -600 \Rightarrow x = 7500

Answer

Emily had

7500

invested in the

3\%

account.

Key Concept

Solving linear equations

Explanation

To find the speed of the current and the amount invested, we set up and solve linear equations based on the given conditions.

Solution by Steps

step 1

Write the system of equations in augmented matrix form:

\left[\begin{array}{ccc|c} 2 &amp; -3 &amp; 3 &amp; 21 \\ 1 &amp; 2 &amp; -1 &amp; 8 \\ -4 &amp; 2 &amp; 1 &amp; -6 \end{array}\right]

step 2

Perform row operations to get a leading 1 in the first row, first column. Divide the first row by 2:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 1 &amp; 2 &amp; -1 &amp; 8 \\ -4 &amp; 2 &amp; 1 &amp; -6 \end{array}\right]

step 3

Subtract the first row from the second row to eliminate the first column in the second row:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 0 &amp; 3.5 &amp; -2.5 &amp; -2.5 \\ -4 &amp; 2 &amp; 1 &amp; -6 \end{array}\right]

step 4

Add 4 times the first row to the third row to eliminate the first column in the third row:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 0 &amp; 3.5 &amp; -2.5 &amp; -2.5 \\ 0 &amp; -4 &amp; 7 &amp; 36 \end{array}\right]

step 5

Divide the second row by 3.5 to get a leading 1 in the second row, second column:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 0 &amp; 1 &amp; -0.714 &amp; -0.714 \\ 0 &amp; -4 &amp; 7 &amp; 36 \end{array}\right]

step 6

Add 4 times the second row to the third row to eliminate the second column in the third row:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 0 &amp; 1 &amp; -0.714 &amp; -0.714 \\ 0 &amp; 0 &amp; 4.144 &amp; 33.144 \end{array}\right]

step 7

Divide the third row by 4.144 to get a leading 1 in the third row, third column:

\left[\begin{array}{ccc|c} 1 &amp; -1.5 &amp; 1.5 &amp; 10.5 \\ 0 &amp; 1 &amp; -0.714 &amp; -0.714 \\ 0 &amp; 0 &amp; 1 &amp; 8 \end{array}\right]

step 8

Substitute back to find the values of

y

and

x

:

\begin{align*} z &amp;= 8 \\ y - 0.714z &amp;= -0.714 \implies y - 5.712 = -0.714 \implies y = 5 \\ x - 1.5y + 1.5z &amp;= 10.5 \implies x - 7.5 + 12 = 10.5 \implies x = 6 \end{align*}

Answer

The solution to the system of equations is

x = 6

,

y = 5

,

z = 8

.

Question 23: What are the dimensions of the matrix?

step 1

Identify the number of rows and columns in the matrix:

\left[\begin{array}{llll} 2 &amp; 9 &amp; 2 &amp; 1 \\ 0 &amp; 7 &amp; 1 &amp; 5 \end{array}\right]

step 2

Count the rows and columns:

\text{Rows} = 2, \quad \text{Columns} = 4

Answer

The dimensions of the matrix are

2 \times 4

.

Key Concept

Gaussian Elimination

Explanation

Gaussian elimination is a method used to solve systems of linear equations by transforming the system's augmented matrix into row-echelon form.

Solution by Steps

step 1

Write the system of equations in matrix form

step 2

The system of equations is:

\begin{aligned} 2x - 3y &amp; = 10 \\ x - 3z &amp; = 3 \\ 4x + 2y + z &amp; = 6 \end{aligned}

step 3

The matrix equation can be written as

A \mathbf{x} = \mathbf{b}

, where

A

is the coefficient matrix,

\mathbf{x}

is the variable matrix, and

\mathbf{b}

is the constant matrix

step 4

The coefficient matrix

A

is:

A = \begin{pmatrix} 2 &amp; -3 &amp; 0 \\ 1 &amp; 0 &amp; -3 \\ 4 &amp; 2 &amp; 1 \end{pmatrix}

step 5

The variable matrix

\mathbf{x}

is:

\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}

step 6

The constant matrix

\mathbf{b}

is:

\mathbf{b} = \begin{pmatrix} 10 \\ 3 \\ 6 \end{pmatrix}

step 7

Therefore, the matrix equation is:

\begin{pmatrix} 2 &amp; -3 &amp; 0 \\ 1 &amp; 0 &amp; -3 \\ 4 &amp; 2 &amp; 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10 \\ 3 \\ 6 \end{pmatrix}

Answer

The matrix equation is: \[ \begin{pmatrix} 2 & -3 & 0 \\ 1 & 0 & -3 \\ 4 & 2 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 10 \\ 3 \\ 6 \end{pmatrix}

Key Concept

Matrix Equation

Explanation

A matrix equation represents a system of linear equations in matrix form, where the coefficient matrix multiplies the variable matrix to yield the constant matrix.

Question 25

step 1

Write the augmented matrix for the system of equations:

\begin{aligned} 2x + y - 3z &amp; = 5 \\ x + 2y + 3z &amp; = 16 \\ -x + 3y - 2z &amp; = 1 \end{aligned}

step 2

The augmented matrix is:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 1 &amp; 2 &amp; 3 &amp; | &amp; 16 \\ -1 &amp; 3 &amp; -2 &amp; | &amp; 1 \end{pmatrix}

step 3

Use row operations to convert the augmented matrix to reduced row echelon form (RREF)

step 4

Perform

R2 - \frac{1}{2}R1 \rightarrow R2

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; \frac{3}{2} &amp; \frac{9}{2} &amp; | &amp; \frac{27}{2} \\ -1 &amp; 3 &amp; -2 &amp; | &amp; 1 \end{pmatrix}

step 5

Perform

R3 + R1 \rightarrow R3

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; \frac{3}{2} &amp; \frac{9}{2} &amp; | &amp; \frac{27}{2} \\ 0 &amp; 4 &amp; -5 &amp; | &amp; 6 \end{pmatrix}

step 6

Perform

\frac{2}{3}R2 \rightarrow R2

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; 1 &amp; 3 &amp; | &amp; 9 \\ 0 &amp; 4 &amp; -5 &amp; | &amp; 6 \end{pmatrix}

step 7

Perform

R3 - 4R2 \rightarrow R3

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; 1 &amp; 3 &amp; | &amp; 9 \\ 0 &amp; 0 &amp; -17 &amp; | &amp; -30 \end{pmatrix}

step 8

Perform

-\frac{1}{17}R3 \rightarrow R3

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; 1 &amp; 3 &amp; | &amp; 9 \\ 0 &amp; 0 &amp; 1 &amp; | &amp; \frac{30}{17} \end{pmatrix}

step 9

Perform

R2 - 3R3 \rightarrow R2

:

\begin{pmatrix} 2 &amp; 1 &amp; -3 &amp; | &amp; 5 \\ 0 &amp; 1 &amp; 0 &amp; | &amp; \frac{3}{17} \\ 0 &amp; 0 &amp; 1 &amp; | &amp; \frac{30}{17} \end{pmatrix}

step 10

Perform

R1 + 3R3 \rightarrow R1

:

\begin{pmatrix} 2 &amp; 1 &amp; 0 &amp; | &amp; \frac{155}{17} \\ 0 &amp; 1 &amp; 0 &amp; | &amp; \frac{3}{17} \\ 0 &amp; 0 &amp; 1 &amp; | &amp; \frac{30}{17} \end{pmatrix}

step 11

Perform

\frac{1}{2}R1 \rightarrow R1

:

\begin{pmatrix} 1 &amp; \frac{1}{2} &amp; 0 &amp; | &amp; \frac{155}{34} \\ 0 &amp; 1 &amp; 0 &amp; | &amp; \frac{3}{17} \\ 0 &amp; 0 &amp; 1 &amp; | &amp; \frac{30}{17} \end{pmatrix}

step 12

Perform

R1 - \frac{1}{2}R2 \rightarrow R1

:

\begin{pmatrix} 1 &amp; 0 &amp; 0 &amp; | &amp; \frac{149}{34} \\ 0 &amp; 1 &amp; 0 &amp; | &amp; \frac{3}{17} \\ 0 &amp; 0 &amp; 1 &amp; | &amp; \frac{30}{17} \end{pmatrix}

step 13

The solution to the system is:

\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \frac{149}{34} \\ \frac{3}{17} \\ \frac{30}{17} \end{pmatrix}

Answer

The solution to the system is: \[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \frac{149}{34} \\ \frac{3}{17} \\ \frac{30}{17} \end{pmatrix}

Key Concept

Gauss-Jordan Elimination

Explanation

Gauss-Jordan elimination is a method for solving linear systems by transforming the augmented matrix to reduced row echelon form.

Question 26

step 1

Write the matrices to be multiplied:

\begin{pmatrix} 1 &amp; 3 \\ 4 &amp; 6 \end{pmatrix} \text{ and } \begin{pmatrix} -3 &amp; 5 &amp; 8 \\ 2 &amp; -1 &amp; 3 \end{pmatrix}

step 2

Perform the matrix multiplication:

\begin{pmatrix} 1 &amp; 3 \\ 4 &amp; 6 \end{pmatrix} \begin{pmatrix} -3 &amp; 5 &amp; 8 \\ 2 &amp; -1 &amp; 3 \end{pmatrix}

step 3

Calculate the elements of the resulting matrix:

\begin{aligned} (1 \cdot -3 + 3 \cdot 2) &amp; = -3 + 6 = 3 \\ (1 \cdot 5 + 3 \cdot -1) &amp; = 5 - 3 = 2 \\ (1 \cdot 8 + 3 \cdot 3) &amp; = 8 + 9 = 17 \\ (4 \cdot -3 + 6 \cdot 2) &amp; = -12 + 12 = 0 \\ (4 \cdot 5 + 6 \cdot -1) &amp; = 20 - 6 = 14 \\ (4 \cdot 8 + 6 \cdot 3) &amp; = 32 + 18 = 50 \end{aligned}

step 4

The resulting matrix is:

\begin{pmatrix} 3 &amp; 2 &amp; 17 \\ 0 &amp; 14 &amp; 50 \end{pmatrix}

Answer

The product of the matrices is: \[ \begin{pmatrix} 3 & 2 & 17 \\ 0 & 14 & 50 \end{pmatrix}

Key Concept

Matrix Multiplication

Explanation

Matrix multiplication involves taking the dot product of rows and columns to produce a new matrix.

]

Solution by Steps

step 1

The given matrices are:

\left[\begin{array}{ll}1 &amp; 9 \\ 3 &amp; 0\end{array}\right]

and

\left[\begin{array}{ll}3 &amp; 7 \\ 2 &amp; 4\end{array}\right]

step 2

Add the corresponding elements of the matrices:

\left[\begin{array}{ll}1+3 &amp; 9+7 \\ 3+2 &amp; 0+4\end{array}\right]

step 3

Simplify the result:

\left[\begin{array}{ll}4 &amp; 16 \\ 5 &amp; 4\end{array}\right]

Answer

\left[\begin{array}{ll}4 &amp; 16 \\ 5 &amp; 4\end{array}\right]

Key Concept

Matrix Addition

Explanation

To add two matrices, add their corresponding elements.

Problem 28: Find the inverse of a matrix

step 1

The given matrix is:

\left[\begin{array}{cc}-1 &amp; 4 \\ 2 &amp; 5\end{array}\right]

step 2

Calculate the determinant:

\text{det} = (-1)(5) - (4)(2) = -5 - 8 = -13

step 3

Use the formula for the inverse of a 2x2 matrix:

\text{Inverse} = \frac{1}{\text{det}} \left[\begin{array}{cc}d &amp; -b \\ -c &amp; a\end{array}\right]

where

a = -1

,

b = 4

,

c = 2

,

d = 5

step 4

Substitute the values and simplify:

\text{Inverse} = \frac{1}{-13} \left[\begin{array}{cc}5 &amp; -4 \\ -2 &amp; -1\end{array}\right] = \left[\begin{array}{cc}-5/13 &amp; 4/13 \\ 2/13 &amp; -1/13\end{array}\right]

Answer

\left[\begin{array}{cc}-5/13 &amp; 4/13 \\ 2/13 &amp; -1/13\end{array}\right]

Key Concept

Matrix Inversion

Explanation

To find the inverse of a 2x2 matrix, use the formula involving the determinant and the elements of the matrix.

Problem 29: Solve the system of equations using an inverse matrix

step 1

The system of equations is:

\left\{\begin{array}{l}2x - 3y = 2 \\ x + 2y = 20\end{array}\right.

step 2

Write the system in matrix form

AX = B

:

A = \left[\begin{array}{cc}2 &amp; -3 \\ 1 &amp; 2\end{array}\right], \quad X = \left[\begin{array}{c}x \\ y\end{array}\right], \quad B = \left[\begin{array}{c}2 \\ 20\end{array}\right]

step 3

Find the inverse of matrix

A

:

A^{-1} = \left[\begin{array}{cc}2 &amp; -3 \\ 1 &amp; 2\end{array}\right]^{-1} = \frac{1}{7} \left[\begin{array}{cc}2 &amp; 3 \\ -1 &amp; 2\end{array}\right] = \left[\begin{array}{cc}2/7 &amp; 3/7 \\ -1/7 &amp; 2/7\end{array}\right]

step 4

Multiply

A^{-1}

by

B

to find

X

:

X = A^{-1}B = \left[\begin{array}{cc}2/7 &amp; 3/7 \\ -1/7 &amp; 2/7\end{array}\right] \left[\begin{array}{c}2 \\ 20\end{array}\right] = \left[\begin{array}{c}(2/7)(2) + (3/7)(20) \\ (-1/7)(2) + (2/7)(20)\end{array}\right] = \left[\begin{array}{c}64/7 \\ 38/7\end{array}\right]

step 5

Simplify the result:

x = \frac{64}{7}, \quad y = \frac{38}{7}

Answer

x = \frac{64}{7}, \quad y = \frac{38}{7}

Key Concept

Solving Systems of Equations Using Inverse Matrices

Explanation

To solve a system of linear equations using an inverse matrix, express the system in matrix form and multiply both sides by the inverse of the coefficient matrix.

}

Solution by Steps

step 1

To find the determinant of matrix

A

, we use the formula for a

2 \times 2

matrix:

|A| = ad - bc

step 2

Here,

a = 9

,

b = 6

,

c = 2

, and

d = 3

step 3

Substitute the values into the formula:

|A| = (9 \cdot 3) - (6 \cdot 2)

step 4

Calculate the determinant:

|A| = 27 - 12 = 15

Answer

15

Key Concept

Determinant of a

2 \times 2

matrix

Explanation

The determinant of a

2 \times 2

matrix is calculated using the formula

|A| = ad - bc

.

Question 31: Set up to solve for

x

using Cramer's Rule?

\left\{\begin{array}{c}x-2 y+3 z=8 \\ 2 x+y-2 z=5 \\ -x-4 y+5 z=2\end{array}\right.

step 1

Write the system of equations in matrix form

AX = B

, where

A

is the coefficient matrix,

X

is the variable matrix, and

B

is the constant matrix

step 2

The coefficient matrix

A

is

\left[\begin{array}{ccc}1 &amp; -2 &amp; 3 \\ 2 &amp; 1 &amp; -2 \\ -1 &amp; -4 &amp; 5\end{array}\right]

step 3

The variable matrix

X

is

\left[\begin{array}{c}x \\ y \\ z\end{array}\right]

step 4

The constant matrix

B

is

\left[\begin{array}{c}8 \\ 5 \\ 2\end{array}\right]

step 5

To solve for

x

using Cramer's Rule, we need to find the determinant of

A

, denoted as

|A|

step 6

Replace the first column of

A

with

B

to form a new matrix

A_x

:

\left[\begin{array}{ccc}8 &amp; -2 &amp; 3 \\ 5 &amp; 1 &amp; -2 \\ 2 &amp; -4 &amp; 5\end{array}\right]

step 7

Calculate the determinant of

A_x

, denoted as

|A_x|

step 8

The value of

x

is given by

x = \frac{|A_x|}{|A|}

Answer

x = \frac{|A_x|}{|A|}

Key Concept

Cramer's Rule

Explanation

Cramer's Rule is used to solve a system of linear equations with as many equations as unknowns, using determinants.

Question 32: What is the partial fraction decomposition of

\frac{x}{2 x^{2}-7 x+3}

?

step 1

Factor the denominator:

2x^2 - 7x + 3 = (2x - 1)(x - 3)

step 2

Write the partial fraction decomposition:

\frac{x}{(2x - 1)(x - 3)} = \frac{A}{2x - 1} + \frac{B}{x - 3}

step 3

Multiply both sides by the denominator

(2x - 1)(x - 3)

to clear the fractions:

x = A(x - 3) + B(2x - 1)

step 4

Expand and collect like terms:

x = Ax - 3A + 2Bx - B

step 5

Combine like terms:

x = (A + 2B)x - (3A + B)

step 6

Equate coefficients of

x

and the constant terms:

A + 2B = 1

and

-3A - B = 0

step 7

Solve the system of equations for

A

and

B

:

A = 1

and

B = -1

step 8

Substitute

A

and

B

back into the partial fractions:

\frac{x}{(2x - 1)(x - 3)} = \frac{1}{2x - 1} - \frac{1}{x - 3}

Answer

\frac{1}{2x - 1} - \frac{1}{x - 3}

Key Concept

Partial Fraction Decomposition

Explanation

Partial fraction decomposition is a method used to express a rational function as a sum of simpler fractions.

}

Generated Graph

Solution by Steps

step 1

Identify the form of the partial fraction decomposition for the given rational function

step 2

The given function is

\frac{3x+1}{(x-1)^2}

step 3

Since the denominator is

(x-1)^2

, we decompose it as

\frac{A}{x-1} + \frac{B}{(x-1)^2}

step 4

Set up the equation:

\frac{3x+1}{(x-1)^2} = \frac{A}{x-1} + \frac{B}{(x-1)^2}

step 5

Multiply both sides by

(x-1)^2

to clear the denominators:

3x+1 = A(x-1) + B

step 6

Expand and combine like terms:

3x+1 = Ax - A + B

step 7

Equate coefficients of like terms:

3x + 1 = Ax - A + B

step 8

From the equation, we get

A = 3

and

-A + B = 1

step 9

Substitute

A = 3

into

-A + B = 1

:

-3 + B = 1

step 10

Solve for

B

:

B = 4

step 11

Therefore, the partial fraction decomposition is

\frac{3}{x-1} + \frac{4}{(x-1)^2}

Answer

\frac{3}{x-1} + \frac{4}{(x-1)^2}

Key Concept

Partial Fraction Decomposition

Explanation

The given rational function is decomposed into simpler fractions whose denominators are factors of the original denominator.

Question 34: Inequalities for the Graphed System

step 1

Identify the shapes and their equations from the graph

step 2

The first shape is a circle centered at the origin with radius

r

step 3

The equation of the circle is

x^2 + y^2 = r^2

step 4

The second shape is an ellipse centered at the origin with horizontal orientation

step 5

The equation of the ellipse is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

step 6

Write the inequalities for the circle and ellipse

step 7

For the circle, the inequality is

x^2 + y^2 \leq r^2

step 8

For the ellipse, the inequality is

\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1

Answer

x^2 + y^2 \leq r^2

and

\frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1

Key Concept

Inequalities for Graphs

Explanation

The inequalities represent the regions inside the circle and ellipse, respectively.

Question 35: Equation of the Parabola

step 1

Identify the vertex and directrix of the parabola

step 2

The vertex is at the origin

(0,0)

and the directrix is

x = -5

step 3

Use the standard form of the equation of a parabola with a vertical axis:

(x-h)^2 = 4p(y-k)

step 4

Since the vertex is at

(0,0)

, the equation simplifies to

x^2 = 4py

step 5

The distance from the vertex to the directrix is

|p| = 5

step 6

Since the directrix is to the left of the vertex,

p = 5

step 7

Substitute

p

into the equation:

x^2 = 4(5)y

step 8

Simplify the equation:

x^2 = 20y

Answer

x^2 = 20y

Key Concept

Equation of a Parabola

Explanation

The equation is derived using the vertex and the distance to the directrix.

Solution by Steps

step 1

The equation of a parabola with vertex at the origin and opening upwards is given by

y = \frac{x^2}{4f}

, where

f

is the focal length

step 2

Given the width at the opening is 12 feet, the distance from the vertex to the edge of the parabola is

6

feet (half of 12 feet)

step 3

The depth of the parabola is given as

5

feet

step 4

Substitute

x = 6

and

y = 5

into the equation

y = \frac{x^2}{4f}

to find

f

:

5 = \frac{6^2}{4f}

5 = \frac{36}{4f}

5 = \frac{9}{f}

f = \frac{9}{5} = 1.8 \text{ feet}

Answer

The bulb should be placed 1.8 feet above the bottom of the reflector.

Question 37

step 1

The standard form of the equation of an ellipse with center at the origin is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

, where

a

is the semi-major axis and

b

is the semi-minor axis

step 2

Given the vertex at

(0,6)

, the length of the semi-major axis

a = 6

step 3

The minor axis length is 2, so the semi-minor axis

b = 1

step 4

Substitute

a

and

b

into the standard form:

\frac{x^2}{6^2} + \frac{y^2}{1^2} = 1

\frac{x^2}{36} + \frac{y^2}{1} = 1

Answer

The equation of the ellipse is

\frac{x^2}{36} + \frac{y^2}{1} = 1

.

Question 38

step 1

The foci of an ellipse are located at

(0, \pm c)

, where

c = \sqrt{a^2 - b^2}

step 2

Given

a = 6

and

b = 1

, calculate

c

:

c = \sqrt{6^2 - 1^2}

c = \sqrt{36 - 1}

c = \sqrt{35}

step 3

The foci are at

(0, \pm \sqrt{35})

Answer

The foci of the ellipse are at

(0, \pm \sqrt{35})

.

Key Concept

Parabola and Ellipse Properties

Explanation

For a parabola, the focal length can be found using the vertex form equation. For an ellipse, the standard form equation helps determine the semi-major and semi-minor axes, and the foci are calculated using the relationship between these axes.

Solution by Steps

step 1

Identify the standard form of the hyperbola equation. The given equation is in the form

\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1

, where

a^2 = 1

and

b^2 = 4

step 2

Determine the center of the hyperbola. Since the equation is not shifted, the center is at

(0, 0)

step 3

Find the vertices. The vertices are at

(0, \pm a)

. Here,

a = 1

, so the vertices are at

(0, \pm 1)

step 4

Calculate the asymptotes. The asymptotes for this hyperbola are given by

y = \pm \frac{a}{b} x

. Substituting

a = 1

and

b = 2

, the asymptotes are

y = \pm \frac{1}{2} x

step 5

Determine the foci. The foci are at

(0, \pm c)

, where

c = \sqrt{a^2 + b^2}

. Here,

c = \sqrt{1 + 4} = \sqrt{5}

, so the foci are at

(0, \pm \sqrt{5})

Answer

Center:

(0, 0)

Vertices:

(0, \pm 1)

Asymptotes:

y = \pm \frac{1}{2} x

Foci:

(0, \pm \sqrt{5})

Key Concept

Hyperbola properties

Explanation

The center, vertices, asymptotes, and foci of a hyperbola can be determined from its standard form equation.

Question 40: Find the equation of the hyperbola with vertices at

(\pm 5,0)

and contains the point

\left(\sqrt{41}, \frac{8}{5}\right)

step 1

Identify the vertices. The vertices are at

(\pm 5, 0)

, so

a = 5

step 2

Write the standard form of the hyperbola equation. Since the vertices are on the x-axis, the equation is

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

. Substituting

a = 5

, we get

\frac{x^2}{25} - \frac{y^2}{b^2} = 1

step 3

Use the given point

\left(\sqrt{41}, \frac{8}{5}\right)

to find

b^2

. Substitute

x = \sqrt{41}

and

y = \frac{8}{5}

into the equation:

\frac{(\sqrt{41})^2}{25} - \frac{\left(\frac{8}{5}\right)^2}{b^2} = 1

step 4

Simplify the equation:

\frac{41}{25} - \frac{\frac{64}{25}}{b^2} = 1

step 5

Solve for

b^2

:

\frac{41}{25} - \frac{64}{25b^2} = 1

. Multiply through by 25:

41 - \frac{64}{b^2} = 25

step 6

Rearrange and solve:

41 - 25 = \frac{64}{b^2}

, so

16 = \frac{64}{b^2}

. Therefore,

b^2 = 4

step 7

Write the final equation of the hyperbola:

\frac{x^2}{25} - \frac{y^2}{4} = 1

Answer

Equation:

\frac{x^2}{25} - \frac{y^2}{4} = 1

Key Concept

Hyperbola equation

Explanation

The equation of a hyperbola can be determined using the vertices and a point on the hyperbola.

Question 41: Where is the center of the shifted conic:

3x^{2} + 6x - 2y^{2} + 8y = 9

step 1

Rewrite the equation in standard form by completing the square. Group the

x

and

y

terms:

3(x^2 + 2x) - 2(y^2 - 4y) = 9

step 2

Complete the square for the

x

-terms:

x^2 + 2x

becomes

(x + 1)^2 - 1

step 3

Complete the square for the

y

-terms:

y^2 - 4y

becomes

(y - 2)^2 - 4

step 4

Substitute back into the equation:

3((x + 1)^2 - 1) - 2((y - 2)^2 - 4) = 9

step 5

Simplify:

3(x + 1)^2 - 3 - 2(y - 2)^2 + 8 = 9

step 6

Combine constants:

3(x + 1)^2 - 2(y - 2)^2 = 4

step 7

Identify the center of the conic. The center is at

(-1, 2)

Answer

Center:

(-1, 2)

Key Concept

Completing the square

Explanation

Completing the square helps to rewrite the conic equation in standard form to identify the center.

Generated Graph

Solution by Steps

step 1

The given equation of the ellipse is

\frac{(x-5)^{2}}{9}+\frac{(y+7)^{2}}{144}=1

step 2

Identify the center of the ellipse:

(h, k) = (5, -7)

step 3

The lengths of the semi-major and semi-minor axes are

a = 12

and

b = 3

, respectively

step 4

Calculate the distance

c

from the center to each focus using

c = \sqrt{a^2 - b^2} = \sqrt{144 - 9} = \sqrt{135} = 3\sqrt{15}

step 5

The foci are located at

(h, k \pm c) = (5, -7 \pm 3\sqrt{15})

step 6

Therefore, the foci are

\left(5, -7 - 3\sqrt{15}\right)

and

\left(5, -7 + 3\sqrt{15}\right)

Answer

The foci of the ellipse are

\left(5, -7 - 3\sqrt{15}\right)

and

\left(5, -7 + 3\sqrt{15}\right)

.

Key Concept

Foci of an ellipse

Explanation

The foci of an ellipse are found using the relationship

c = \sqrt{a^2 - b^2}

, where

a

and

b

are the lengths of the semi-major and semi-minor axes, respectively.

Question 43: Evaluate the sum of

2M+k

from

k=8

to

12

step 1

The given sum is

\sum_{k=8}^{12} (2M + k)

step 2

Separate the sum into two parts:

\sum_{k=8}^{12} 2M + \sum_{k=8}^{12} k

step 3

Evaluate the first part:

\sum_{k=8}^{12} 2M = 2M \cdot 5 = 10M

step 4

Evaluate the second part:

\sum_{k=8}^{12} k = 8 + 9 + 10 + 11 + 12 = 50

step 5

Combine the results:

10M + 50

Answer

The sum of

2M+k

from

k=8

to

12

is

10M + 50

.

Key Concept

Summation of a series

Explanation

The sum of a series can be separated into simpler parts and evaluated individually before combining the results.

Question 44: Find the 7th term of the sequence

\frac{3}{4}, \frac{7}{6}, \frac{11}{8}, \ldots

step 1

Identify the pattern in the sequence. The numerators increase by 4:

3, 7, 11, \ldots

step 2

The general form for the numerator is

3 + 4(n-1) = 4n - 1

step 3

The denominators increase by 2:

4, 6, 8, \ldots

step 4

The general form for the denominator is

4 + 2(n-1) = 2n + 2

step 5

For the 7th term, substitute

n = 7

: numerator =

4(7) - 1 = 27

, denominator =

2(7) + 2 = 16

step 6

Therefore, the 7th term is

\frac{27}{16}

Answer

The 7th term of the sequence is

\frac{27}{16}

.

Key Concept

Arithmetic sequence

Explanation

The terms of the sequence follow a linear pattern in both the numerator and the denominator, which can be expressed in general forms and used to find any term in the sequence.

Solution by Steps

step 1

Recognize the pattern in the number of oranges per level. The sequence is 1, 5, 9, ..., which is an arithmetic sequence with the first term

a_1 = 1

and common difference

d = 4

step 2

Use the formula for the sum of the first

n

terms of an arithmetic sequence:

S_n = \frac{n}{2} (2a_1 + (n-1)d)

. Here,

n = 24

,

a_1 = 1

, and

d = 4

step 3

Substitute the values into the formula:

S_{24} = \frac{24}{2} (2 \cdot 1 + (24-1) \cdot 4)

step 4

Simplify inside the parentheses:

S_{24} = 12 (2 + 23 \cdot 4) = 12 (2 + 92) = 12 \cdot 94

step 5

Calculate the final sum:

S_{24} = 12 \cdot 94 = 1128

Answer

1128 oranges are needed for 24 levels.

Key Concept

Arithmetic Sequence Sum

Explanation

The sum of the first

n

terms of an arithmetic sequence can be found using the formula

S_n = \frac{n}{2} (2a_1 + (n-1)d)

.

Question 46: Find the 200th term of

5, 12, 19, 26, \ldots

step 1

Recognize the pattern in the sequence. The sequence is an arithmetic sequence with the first term

a_1 = 5

and common difference

d = 7

step 2

Use the formula for the

n

-th term of an arithmetic sequence:

a_n = a_1 + (n-1)d

. Here,

n = 200

,

a_1 = 5

, and

d = 7

step 3

Substitute the values into the formula:

a_{200} = 5 + (200-1) \cdot 7

step 4

Simplify the expression:

a_{200} = 5 + 199 \cdot 7 = 5 + 1393

step 5

Calculate the final term:

a_{200} = 1398

Answer

The 200th term is 1398.

Key Concept

Arithmetic Sequence Term

Explanation

The

n

-th term of an arithmetic sequence can be found using the formula

a_n = a_1 + (n-1)d

.

Question 47: Find the sum of the first 10 terms of the sequence

a_n = 200 \left(\frac{5}{7}\right)^{n-1}

step 1

Recognize the pattern in the sequence. The sequence is a geometric sequence with the first term

a_1 = 200

and common ratio

r = \frac{5}{7}

step 2

Use the formula for the sum of the first

n

terms of a geometric sequence:

S_n = a_1 \frac{1-r^n}{1-r}

. Here,

n = 10

,

a_1 = 200

, and

r = \frac{5}{7}

step 3

Substitute the values into the formula:

S_{10} = 200 \frac{1 - \left(\frac{5}{7}\right)^{10}}{1 - \frac{5}{7}}

step 4

Simplify the denominator:

1 - \frac{5}{7} = \frac{2}{7}

step 5

Substitute and simplify:

S_{10} = 200 \frac{1 - \left(\frac{5}{7}\right)^{10}}{\frac{2}{7}} = 200 \cdot \frac{7}{2} \left(1 - \left(\frac{5}{7}\right)^{10}\right)

step 6

Calculate the final sum:

S_{10} = 700 \left(1 - \left(\frac{5}{7}\right)^{10}\right)

Answer

The sum of the first 10 terms is

700 \left(1 - \left(\frac{5}{7}\right)^{10}\right)

.

Key Concept

Geometric Sequence Sum

Explanation

The sum of the first

n

terms of a geometric sequence can be found using the formula

S_n = a_1 \frac{1-r^n}{1-r}

.

Solution by Steps

step 1

Identify the first term (

a

) and the common ratio (

r

) of the geometric series. The first term is

a = \frac{8}{3}

step 2

To find the common ratio

r

, divide the second term by the first term:

r = \frac{\frac{40}{12}}{\frac{8}{3}} = \frac{40}{12} \cdot \frac{3}{8} = \frac{40 \cdot 3}{12 \cdot 8} = \frac{120}{96} = \frac{5}{4}

step 3

Verify that the series is geometric by checking the ratio between subsequent terms. For example,

\frac{\frac{200}{48}}{\frac{40}{12}} = \frac{200 \cdot 12}{48 \cdot 40} = \frac{2400}{1920} = \frac{5}{4}

, confirming

r = \frac{5}{4}

step 4

Since |r| > 1, the series diverges. Therefore, the sum of the infinite geometric series does not exist

Answer

The series diverges

Key Concept

Divergence of geometric series

Explanation

For a geometric series to converge, the absolute value of the common ratio must be less than 1. In this case, |r| = \frac{5}{4} > 1, so the series diverges.

Solution by Steps

step 1

Identify the given values: Britney saves $100 per week, the interest rate is 8% per year compounded weekly, and the duration is 17 years

step 2

Convert the annual interest rate to a weekly interest rate:

r = \frac{0.08}{52}

step 3

Calculate the total number of weeks:

n = 17 \times 52

step 4

Use the future value of an annuity formula:

FV = P \left( \frac{(1 + r)^n - 1}{r} \right)

, where

P = 100

,

r = \frac{0.08}{52}

, and

n = 17 \times 52

step 5

Substitute the values into the formula and calculate:

FV = 100 \left( \frac{(1 + \frac{0.08}{52})^{884} - 1}{\frac{0.08}{52}} \right)

step 6

Simplify and compute the final amount

Answer

FV \approx 193,964.24

Key Concept

Future Value of an Annuity

Explanation

This formula calculates the future value of regular savings with compound interest.

Question 50

step 1

Identify the given values: Loan amount

PV = 100,000

, annual interest rate

r = 0.045

, and loan term

n = 15

years with monthly payments

step 2

Convert the annual interest rate to a monthly interest rate:

r_{monthly} = \frac{0.045}{12}

step 3

Calculate the total number of payments:

n_{monthly} = 15 \times 12

step 4

Use the loan payment formula:

PMT = PV \left( \frac{r_{monthly} (1 + r_{monthly})^{n_{monthly}}}{(1 + r_{monthly})^{n_{monthly}} - 1} \right)

step 5

Substitute the values into the formula and calculate:

PMT = 100,000 \left( \frac{\frac{0.045}{12} (1 + \frac{0.045}{12})^{180}}{(1 + \frac{0.045}{12})^{180} - 1} \right)

step 6

Simplify and compute the monthly payment

Answer

PMT \approx 764.99

Key Concept

Loan Payment Calculation

Explanation

This formula calculates the monthly payment required to pay off a loan with compound interest over a specified term.

Question 51

step 1

State the base case: For

n = 1

, the left-hand side is

4 + 6 + 8 + 2(1) + 2 = 4 + 6 + 8 + 2 + 2 = 22

, and the right-hand side is

1(1 + 3) = 4

step 2

Clearly, the base case does not hold. Recheck the problem statement or initial conditions

step 3

Assume the statement is true for

n = k

:

4 + 6 + 8 + 2k + 2 = k(k + 3)

step 4

Prove for

n = k + 1

:

4 + 6 + 8 + 2(k + 1) + 2 = (k + 1)((k + 1) + 3)

step 5

Simplify the left-hand side:

4 + 6 + 8 + 2k + 2 + 2 = k(k + 3) + 2(k + 1) + 2

step 6

Simplify the right-hand side:

(k + 1)(k + 4) = k^2 + 5k + 4

step 7

Verify both sides are equal:

k^2 + 3k + 2 + 2k + 2 = k^2 + 5k + 4

Answer

The statement is proven by induction.

Key Concept

Mathematical Induction

Explanation

This method proves a statement is true for all natural numbers by proving it for the base case and then assuming it for

n = k

to prove it for

n = k + 1

.

Generated Graph

Solution by Steps

step 1

To find the 3rd term in the expansion of

(2x - 3y)^7

, we use the binomial theorem:

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

step 2

Here,

a = 2x

,

b = -3y

, and

n = 7

. The 3rd term corresponds to

k = 2

step 3

The 3rd term is given by

\binom{7}{2} (2x)^{7-2} (-3y)^2

step 4

Calculate

\binom{7}{2} = \frac{7!}{2!(7-2)!} = 21

step 5

Calculate

(2x)^5 = 32x^5

and

(-3y)^2 = 9y^2

step 6

Multiply these together:

21 \cdot 32x^5 \cdot 9y^2 = 6048x^5y^2

Answer

The 3rd term in the expansion of

(2x - 3y)^7

is

6048x^5y^2

.

Key Concept

Binomial Theorem

Explanation

The binomial theorem allows us to expand expressions of the form

(a + b)^n

and find specific terms in the expansion.

Question 53: What is the coefficient of the

y^2

term of

(2x - 3y)^7

step 1

To find the coefficient of the

y^2

term in the expansion of

(2x - 3y)^7

, we use the binomial theorem:

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

step 2

Here,

a = 2x

,

b = -3y

, and

n = 7

. The

y^2

term corresponds to

k = 2

step 3

The term is given by

\binom{7}{2} (2x)^{7-2} (-3y)^2

step 4

Calculate

\binom{7}{2} = \frac{7!}{2!(7-2)!} = 21

step 5

Calculate

(2x)^5 = 32x^5

and

(-3y)^2 = 9y^2

step 6

The coefficient is

21 \cdot 32 \cdot 9 = 6048

Answer

The coefficient of the

y^2

term in the expansion of

(2x - 3y)^7

is

6048

.

Key Concept

Binomial Coefficient

Explanation

The binomial coefficient

\binom{n}{k}

determines the coefficients in the expansion of

(a + b)^n

.

Question 54: Find

\lim_{x \rightarrow 0} \frac{x^3 - 27}{x^3 - 2x - 3}

step 1

To find the limit

\lim_{x \rightarrow 0} \frac{x^3 - 27}{x^3 - 2x - 3}

, we substitute

x = 0

directly into the expression

step 2

Substituting

x = 0

gives

\frac{0^3 - 27}{0^3 - 2(0) - 3} = \frac{-27}{-3} = 9

Answer

The limit

\lim_{x \rightarrow 0} \frac{x^3 - 27}{x^3 - 2x - 3}

is

9

.

Key Concept

Direct Substitution

Explanation

When finding limits, direct substitution can be used if the function is continuous at the point of interest.

Generated Graph

Solution by Steps

step 1

We start by simplifying the given limit expression:

\lim_{x \to 3} \frac{x^3 - 27}{x^2 - 2x - 3}

step 2

Notice that

x^3 - 27

can be factored as

(x - 3)(x^2 + 3x + 9)

and

x^2 - 2x - 3

can be factored as

(x - 3)(x + 1)

step 3

Substitute these factorizations into the limit expression:

\lim_{x \to 3} \frac{(x - 3)(x^2 + 3x + 9)}{(x - 3)(x + 1)}

step 4

Cancel the common factor

(x - 3)

in the numerator and the denominator:

\lim_{x \to 3} \frac{x^2 + 3x + 9}{x + 1}

step 5

Evaluate the limit by substituting

x = 3

:

\frac{3^2 + 3 \cdot 3 + 9}{3 + 1} = \frac{9 + 9 + 9}{4} = \frac{27}{4}

Answer

\frac{27}{4}

Key Concept

Limit of a rational function

Explanation

To find the limit of a rational function as

x

approaches a certain value, factor the numerator and the denominator, cancel common factors, and then substitute the value of

x

.

Generated Graph

Solution by Steps

step 1

To find the limit

\lim_{x \to 3} \frac{x-3}{x-5}

, we first observe that direct substitution of

x = 3

into the expression

\frac{x-3}{x-5}

results in

\frac{0}{-2}

, which simplifies to

0

step 2

Therefore,

\lim_{x \to 3} \frac{x-3}{x-5} = 0

Answer

0

Key Concept

Limit of a function

Explanation

The limit of a function as

x

approaches a certain value can often be found by direct substitution, provided it does not result in an indeterminate form.

Solution by Steps

step 1

To find the limit

\lim_{x \to 2} \frac{\sqrt{x+2}-2}{x-2}

, we first observe that direct substitution of

x = 2

into the expression

\frac{\sqrt{x+2}-2}{x-2}

results in

\frac{0}{0}

, which is an indeterminate form

step 2

To resolve this, we can use the technique of multiplying by the conjugate. Multiply the numerator and the denominator by

\sqrt{x+2} + 2

:

step 3

\frac{\sqrt{x+2}-2}{x-2} \cdot \frac{\sqrt{x+2}+2}{\sqrt{x+2}+2} = \frac{(\sqrt{x+2})^2 - 2^2}{(x-2)(\sqrt{x+2}+2)} = \frac{x+2-4}{(x-2)(\sqrt{x+2}+2)} = \frac{x-2}{(x-2)(\sqrt{x+2}+2)}

step 4

Simplify the expression:

\frac{x-2}{(x-2)(\sqrt{x+2}+2)} = \frac{1}{\sqrt{x+2}+2}

step 5

Now, substitute

x = 2

into the simplified expression:

\frac{1}{\sqrt{2+2}+2} = \frac{1}{2+2} = \frac{1}{4}

Answer

\frac{1}{4}

Key Concept

Limit involving square roots

Explanation

When dealing with limits that result in an indeterminate form, multiplying by the conjugate can help simplify the expression to find the limit.

Solution by Steps

step 1

To find the derivative of

f(x) = 2x^2 + 5

at

x = 3

, we first find the general derivative of the function

step 2

The derivative of

f(x) = 2x^2 + 5

is found using the power rule:

\frac{d}{dx}(2x^2 + 5) = 4x

step 3

Now, substitute

x = 3

into the derivative:

4(3) = 12

Answer

12

Key Concept

Derivative using the power rule

Explanation

The power rule states that the derivative of

x^n

is

nx^{n-1}

. This rule is applied to each term of the polynomial to find the derivative.

Generated Graph

Solution by Steps

step 1

To find the 3rd term in the expansion of

(2x - 3y)^7

, we use the binomial theorem:

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

step 2

Here,

a = 2x

,

b = -3y

, and

n = 7

. The 3rd term corresponds to

k = 2

step 3

The 3rd term is given by

\binom{7}{2} (2x)^{7-2} (-3y)^2

step 4

Calculate

\binom{7}{2} = \frac{7!}{2!(7-2)!} = 21

step 5

Calculate

(2x)^5 = 32x^5

and

(-3y)^2 = 9y^2

step 6

Multiply these together:

21 \cdot 32x^5 \cdot 9y^2 = 6048x^5y^2

Answer

The 3rd term in the expansion of

(2x - 3y)^7

is

6048x^5y^2

.

Key Concept

Binomial Theorem

Explanation

The binomial theorem allows us to expand expressions of the form

(a + b)^n

and find specific terms in the expansion.

Question 53: What is the coefficient of the

y^2

term of

(2x - 3y)^7

step 1

To find the coefficient of the

y^2

term in the expansion of

(2x - 3y)^7

, we use the binomial theorem:

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

step 2

Here,

a = 2x

,

b = -3y

, and

n = 7

. The

y^2

term corresponds to

k = 2

step 3

The coefficient of the

y^2

term is given by

\binom{7}{2} (2x)^{7-2} (-3y)^2

step 4

Calculate

\binom{7}{2} = \frac{7!}{2!(7-2)!} = 21

step 5

Calculate

(2x)^5 = 32x^5

and

(-3y)^2 = 9y^2

step 6

Multiply these together:

21 \cdot 32x^5 \cdot 9y^2 = 6048x^5y^2

step 7

The coefficient of the

y^2

term is

6048x^5

Answer

The coefficient of the

y^2

term in the expansion of

(2x - 3y)^7

is

6048x^5

.

Key Concept

Binomial Coefficient

Explanation

The binomial coefficient

\binom{n}{k}

determines the coefficient of each term in the binomial expansion.

Question 54: Find

\lim_{x \rightarrow 3} \frac{x^3 - 27}{x^2 - 2x - 3}

step 1

To find the limit

\lim_{x \rightarrow 3} \frac{x^3 - 27}{x^2 - 2x - 3}

, we first factorize the numerator and the denominator

step 2

The numerator

x^3 - 27

can be factored as

(x - 3)(x^2 + 3x + 9)

step 3

The denominator

x^2 - 2x - 3

can be factored as

(x - 3)(x + 1)

step 4

The limit becomes

\lim_{x \rightarrow 3} \frac{(x - 3)(x^2 + 3x + 9)}{(x - 3)(x + 1)}

step 5

Cancel the common factor

(x - 3)

:

\lim_{x \rightarrow 3} \frac{x^2 + 3x + 9}{x + 1}

step 6

Substitute

x = 3

:

\frac{3^2 + 3 \cdot 3 + 9}{3 + 1} = \frac{9 + 9 + 9}{4} = \frac{27}{4}

Answer

The limit is

\frac{27}{4}

.

Key Concept

Limit Calculation

Explanation

Factoring the numerator and denominator allows us to simplify the expression and find the limit.

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