Jackie makes an $800 deposit into a bank account earning 4.5% interest.
a. If the interest is compounded quarterly how much money will Jackie have in 10 years?
b. If the interest is compounded continuously how much money will Jackie have in 10 years?
C. If Jackie's money is compounded continuously, how long will it take her money to double?
d. If Jackie's money is compounded monthly, how long will it take her money to triple?

For part a, Jackie will have approximately $1217.62 after 10 years with quarterly compounding interest. For part b, Jackie will have approximately $1226.96 after 10 years with continuous compounding interest. For part c, it will take approximately 15.43 years for Jackie's money to double with continuous compounding interest. For part d, it will take approximately 38.51 years for Jackie's money to triple with monthly compounding interest.

For part a, to calculate the amount of money Jackie will have in 10 years with quarterly compounding interest, we can use the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times compounded per year, and t is the number of years. Plugging in the values, we have A = 800(1 + 0.045/4)^(4*10), which gives us A ≈ $1217.62.

For part b, to calculate the amount of money Jackie will have in 10 years with continuous compounding interest, we can use the formula: A = P * e^(rt), where e is Euler's number and approximately equal to 2.71828. Plugging in the values, we have A = 800 * e^(0.045 * 10), which gives us A ≈ $1226.96.

For part c, to find out how long it will take Jackie's money to double with continuous compounding interest, we can use the formula: t = ln(2) / r, where t is the time in years and r is the interest rate. Plugging in the values, we have t = ln(2) / 0.045, which gives us t ≈ 15.43 years.

For part d, to find out how long it will take Jackie's money to triple with monthly compounding interest, we can use the formula: t = ln(3) / (n * ln(1 + r/n)), where t is the time in years, n is the number of times compounded per year, and r is the interest rate. Plugging in the values, we have t = ln(3) / (12 * ln(1 + 0.045/12)), which gives us t ≈ 38.51 years.

a. If the interest is compounded quarterly, Jackie will have approximately $1,313.63 in 10 years.

b. If the interest is compounded continuously, Jackie will have approximately $1,256.39 in 10 years.

c. If Jackie's money is compounded continuously, it will take approximately 15.4 years for her money to double.

d. If Jackie's money is compounded monthly, it will take approximately 62.47 years for her money to triple.

Step 1

To solve this problem, we can use the compound interest formula:

[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]

Where:

- A is the amount of money accumulated after t years, including interest.

- P is the principal amount (the initial deposit).

- r is the annual interest rate (as a decimal).

- n is the number of times interest is compounded per year.

- t is the time the money is invested for, in years.

Given:

- [tex]\( P = \$800 \)[/tex]

-[tex]\( r = 0.045 \)[/tex] (4.5% as a decimal)

- n is either 4 for quarterly compounding or tends to infinity for continuous compounding.

Step 2

We'll calculate each part step by step:

a. Quarterly compounding:

- [tex]\( n = 4 \)[/tex]

- [tex]\( t = 10 \)[/tex]

Substituting into the formula:

[tex]\[ A = 800 \left(1 + \frac{0.045}{4}\right)^{4 \times 10} \]\[ A = 800 \left(1 + \frac{0.045}{4}\right)^{40} \]\[ A = 800 \left(1 + 0.01125\right)^{40} \]\[ A = 800 \times 1.01125^{40} \]\[ A = 800 \times 1.642032881 \]\[ A = \$1,313.63 \][/tex]

Step 3

b. Continuous compounding:

For continuous compounding, we use the formula:

[tex]\[ A = P e^{rt} \][/tex]

Where e is the base of the natural logarithm, approximately equal to 2.71828.

Substituting the values:

[tex]\[ A = 800 \times e^{0.045 \times 10} \]\[ A = 800 \times e^{0.45} \]\[ A = 800 \times 1.570482165 \]\[ A = \$1,256.39 \][/tex]

Step 4

c. To find out how long it takes for the money to double with continuous compounding, we use the formula:

[tex]\[ A = P e^{rt} \][/tex]

We need to solve for t  when [tex]\( A = 2P \):[/tex]

[tex]\[ 2P = P e^{0.045t} \]\[ 2 = e^{0.045t} \]\[ \ln(2) = 0.045t \]\[ t = \frac{\ln(2)}{0.045} \][/tex]

Substituting the values:

[tex]\[ t = \frac{\ln(2)}{0.045} \]\[ t = \frac{0.693}{0.045} \]\[ t = 15.4 \text{ years} \][/tex]

So, it takes approximately 15.4 years for Jackie's money to double with continuous compounding.

Step 5

d. To find out how long it takes for the money to triple with monthly compounding, we use the compound interest formula again, but this time we set [tex]\( A = 3P \)[/tex]:

[tex]\[ 3P = P \left(1 + \frac{0.045}{12}\right)^{12t} \]\[ 3 = \left(1 + \frac{0.045}{12}\right)^{12t} \]\[ \ln(3) = 12t \ln\left(1 + \frac{0.045}{12}\right) \]\[ t = \frac{\ln(3)}{12 \ln\left(1 + \frac{0.045}{12}\right)} \][/tex]

Substituting the values:

[tex]\[ t = \frac{\ln(3)}{12 \ln\left(1 + \frac{0.045}{12}\right)} \]\[ t = \frac{1.099}{12 \times 0.0037} \]\[ t = 62.47 \text{ years} \][/tex]

So, it takes approximately 62.47 years for Jackie's money to triple with monthly compounding.

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