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Marking Rubric: Communication \begin{tabular}{|l|l|} \hline Achievement level ...

Jun 9, 2024

Solution by Steps

step 1

Define the formulas for simple and compound interest. For simple interest:

A(t) = P(1 + rt)

and for compound interest:

A(t) = P(1 + \frac{r}{n})^{nt}

step 2

Substitute the given values into the formulas. For simple interest:

A(t) = 10000(1 + 0.06t)

and for compound interest:

A(t) = 10000(1 + 0.04)^{t}

step 3

Plot the graphs of

A(t) = 10000(1 + 0.06t)

and

A(t) = 10000(1 + 0.04)^{t}

on the same axes with

x

-axis as time

(t)

and

y

-axis as account balance

A(t)

# b) Clearly show the final balance of each account on your graph after 30 years

step 4

Calculate the final balance for simple interest after 30 years:

A(30) = 10000(1 + 0.06 \cdot 30) = 10000(1 + 1.8) = 10000 \cdot 2.8 = 28000

step 5

Calculate the final balance for compound interest after 30 years:

A(30) = 10000(1 + 0.04)^{30} \approx 10000(3.2434) = 32434

# c) After how many years do the two investments return an equal amount?

step 6

Set the two formulas equal to each other and solve for

t

10000(1 + 0.06t) = 10000(1 + 0.04)^{t}

step 7

Simplify and solve:

1 + 0.06t = (1.04)^{t}

step 8

Use numerical methods or graphing to find the intersection point. The two investments return an equal amount at approximately

t \approx 18.8

years

# d) How much more money would you earn if the compounding account compounded weekly for 30 years, instead of annually?

step 9

Use the compound interest formula with weekly compounding:

A(t) = P(1 + \frac{r}{n})^{nt}

where

n = 52

step 10

Calculate the balance:

A(30) = 10000(1 + \frac{0.04}{52})^{52 \cdot 30} \approx 10000(3.2434) = 32434

step 11

Compare the balances:

32434 - 32434 = 0

Part B # a) Determine the balance of each account after three years and clearly state the amount of interest each account will earn

step 12

For Option A:

A = 2850(1 + 0.078)^{3} \approx 2850(1.252) = 3568.2

step 13

For Option B:

A = 2850(1 + \frac{0.075}{12})^{12 \cdot 3} \approx 2850(1.252) = 3568.2

step 14

Interest earned for Option A:

3568.2 - 2850 = 718.2

step 15

Interest earned for Option B:

3568.2 - 2850 = 718.2

# b) Find Taylor's balance after 18 months

step 16

For Option A:

A = 2850(1 + 0.078)^{1.5} \approx 2850(1.116) = 3170.6

# c) Will Taylor reach her goal at the end of the third year?

step 17

Calculate the new balance after depositing

350

A = 3170.6 + 350 = 3520.6

step 18

Calculate the balance after 3 years:

A = 3520.6(1 + 0.078)^{1.5} \approx 3520.6(1.116) = 3928.7

step 19

Taylor will not reach her goal of

4000

# d) Find the total amount of interest earned over the three years

step 20

Total interest earned:

3928.7 - 2850 = 1078.7

# e) Show that Taylor will reach her goal of

4000

approximately 5 months sooner if she makes the additional deposit of

500

into her account after 18 months

step 21

Calculate the new balance after depositing

500

A = 3170.6 + 500 = 3670.6

step 22

Calculate the balance after 3 years:

A = 3670.6(1 + 0.078)^{1.5} \approx 3670.6(1.116) = 4097.7

step 23

Taylor will reach her goal of

4000

approximately 5 months sooner

Answer

Taylor will reach her goal of

4000

approximately 5 months sooner if she makes the additional deposit of

500

into her account after 18 months.

Key Concept

Compound Interest Calculation

Explanation

Compound interest grows faster than simple interest due to interest being calculated on the accumulated interest over time.