CALCU-The height and radius of a right circular cy

The height and radius of a right circular cylinder are equal, so the cylinder's volume is V = πh³. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h.

Generated Graph

Solution by Steps

step 1

To find the derivative of the volume with respect to the height, we calculate

\frac{d(\pi h^3)}{dh}

step 2

Using the power rule for differentiation, we get

\frac{d}{dh}(\pi h^3) = 3\pi h^2

step 3

The volume error tolerance is 1% of the true value, so we set up the equation

0.01\pi h^3 = 3\pi h^2 dh

step 4

To solve for

dh

, we divide both sides by

3\pi h^2

, yielding

dh = \frac{0.01\pi h^3}{3\pi h^2}

step 5

Simplifying the equation, we get

dh = \frac{0.01 h}{3}

step 6

To express

dh

as a percentage of

h

, we multiply by 100, resulting in

dh/h = \frac{0.01}{3} \times 100

step 7

Calculating the percentage, we find

dh/h = \frac{1}{3}

%

Answer

The greatest error that can be tolerated in the measurement of

h

, expressed as a percentage of

h

, is approximately

\frac{1}{3}

%

Key Concept

Differentiation and error tolerance in measurements

Explanation

The key concept is to use the derivative of the volume with respect to the height to find the relationship between the change in height (

dh

) and the change in volume (

dV

). By setting the maximum volume error to 1% of the volume, we can solve for the maximum allowable error in the height measurement as a percentage of

h

.

I'm asking for the error greatest error that can be tolerated i

Solution by Steps

step 1

To find the greatest error that can be tolerated in the measurement of

h

, we first need to differentiate the volume formula

V = \pi h^3

with respect to

h

step 2

Differentiating

V

with respect to

h

gives us

\frac{dV}{dh} = 3\pi h^2

step 3

The error in

V

, denoted as

dV

, can be approximated by

dV \approx \frac{dV}{dh} dh

, where

dh

is the error in

h

step 4

We want the error

dV

to be no more than 1% of the true volume

V

, so

|dV| \leq 0.01V

step 5

Substituting

V = \pi h^3

and

\frac{dV}{dh} = 3\pi h^2

into the inequality

|dV| \leq 0.01V

, we get

|3\pi h^2 dh| \leq 0.01\pi h^3

step 6

Dividing both sides of the inequality by

3\pi h^2

, we find

|dh| \leq \frac{0.01\pi h^3}{3\pi h^2}

step 7

Simplifying the inequality gives

|dh| \leq \frac{0.01h}{3}

step 8

Expressing

dh

as a percentage of

h

, we get

\frac{|dh|}{h} \leq \frac{0.01}{3}

, which is approximately

\frac{1}{3} \%

Answer

The greatest error that can be tolerated in the measurement of

h

, expressed as a percentage of

h

, is approximately

\frac{1}{3} \%

Key Concept

Differential approximation of errors

Explanation

The greatest tolerable error in the measurement of a quantity can be found by differentiating the function that relates the quantity to its measurement, and then using the differential to approximate the maximum error as a percentage of the true value.

Let

\mathrm{r}

and

\mathrm{s}

be the two roots of

x^{2}-3 x+1

. Find, with proof,

r^{10}+s^{10}

. (Hint: in one approach, Vieta is coupled with Recursion.)

Generated Graph

Solution by Steps

step 1

Given the quadratic equation

x^2 - 3x + 1 = 0

, we find its roots using the quadratic formula

step 2

The roots are

x = \frac{3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}

, which simplifies to

x = \frac{3 \pm \sqrt{5}}{2}

step 3

Let

r = \frac{3 - \sqrt{5}}{2}

and

s = \frac{3 + \sqrt{5}}{2}

be the two roots of the equation

step 4

By Vieta's formulas,

r + s = 3

and

rs = 1

step 5

To find

r^{10} + s^{10}

, we can use the binomial theorem or recognize it as a sum of even powers, which can be expressed in terms of

r + s

and

rs

step 6

We can express

r^{10} + s^{10}

recursively in terms of lower powers of

r

and

s

using the relations

r + s = 3

and

rs = 1

step 7

We find that

r^2 + s^2 = (r + s)^2 - 2rs = 3^2 - 2 = 7

step 8

Similarly,

r^4 + s^4 = (r^2 + s^2)^2 - 2(r^2s^2) = 7^2 - 2 = 47

step 9

Continuing this process, we can find

r^8 + s^8

and then use it to find

r^{10} + s^{10}

step 10

Finally,

r^{10} + s^{10} = (r^5 + s^5)^2 - 2(r^5s^5)

. Since

r^5s^5 = (rs)^5 = 1

, we only need to find

r^5 + s^5

step 11

Using the recursive relation, we find

r^5 + s^5 = (r + s)(r^4 + s^4) - rs(r^3 + s^3)

. We already have

r + s

and

r^4 + s^4

, and we can find

r^3 + s^3

similarly

step 12

After calculating

r^3 + s^3

and using it in the previous step, we can find

r^5 + s^5

and then

r^{10} + s^{10}

Answer

[Insert final answer here after calculating

r^3 + s^3

,

r^5 + s^5

, and

r^{10} + s^{10}

]

Key Concept

Recursion and Vieta's Formulas

Explanation

The sum of even powers of the roots of a quadratic equation can be found using recursive relations derived from Vieta's formulas, which relate the sum and product of the roots to the coefficients of the equation.

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