QuestionQuestion 4 $\$ 7000$ are invested in a bank account at an interest rate of 6 percent per year. Find the amount in the bank after 10 years if interest is compounded annually. $\square$ Find the amount in the bank after 10 years if interest is compounded quarterly. $\square$ Find the amount in the bank after 10 years if interest is compounded monthly. $\square$ Finally, find the amount in the bank after 10 years if interest is compounded contin $\square$ Question Help: Video Message instructor

Studdy Solution

STEP 1

1. The principal amount invested is \$7000.
2. The annual interest rate is 6%.
3. We are calculating the amount after 10 years.
4. Interest is compounded in different frequencies: annually, quarterly, monthly, and continuously.

STEP 2

1. Calculate the amount with annual compounding.
2. Calculate the amount with quarterly compounding.
3. Calculate the amount with monthly compounding.
4. Calculate the amount with continuous compounding.

STEP 3

Calculate the amount with annual compounding.
The formula for compound interest is:
$A = P \left(1 + \frac{r}{n}\right)^{nt}$
Where: - $A$ is the amount of money accumulated after n years, including interest. - $P$ is the principal amount (\$7000). - \( r \) is the annual interest rate (decimal) (0.06). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years (10).
For annual compounding, $n = 1$ .
$A = 7000 \left(1 + \frac{0.06}{1}\right)^{1 \times 10}$ $A = 7000 \times (1.06)^{10}$
Calculate $A$ :
$A \approx 7000 \times 1.790847$ $A \approx 12535.93$
The amount after 10 years with annual compounding is approximately \$12,535.93.

STEP 4

Calculate the amount with quarterly compounding.
For quarterly compounding, $n = 4$ .
$A = 7000 \left(1 + \frac{0.06}{4}\right)^{4 \times 10}$ $A = 7000 \times \left(1 + 0.015\right)^{40}$ $A = 7000 \times (1.015)^{40}$
Calculate $A$ :
$A \approx 7000 \times 1.814019$ $A \approx 12698.13$
The amount after 10 years with quarterly compounding is approximately \$12,698.13.

STEP 5

Calculate the amount with monthly compounding.
For monthly compounding, $n = 12$ .
$A = 7000 \left(1 + \frac{0.06}{12}\right)^{12 \times 10}$ $A = 7000 \times \left(1 + 0.005\right)^{120}$ $A = 7000 \times (1.005)^{120}$
Calculate $A$ :
$A \approx 7000 \times 1.819396$ $A \approx 12735.77$
The amount after 10 years with monthly compounding is approximately \$12,735.77.

STEP 6

Calculate the amount with continuous compounding.
The formula for continuous compounding is:
$A = Pe^{rt}$
Where: - $e$ is the base of the natural logarithm. - $r$ is the annual interest rate (decimal) (0.06). - $t$ is the time the money is invested for in years (10).
$A = 7000 \times e^{0.06 \times 10}$ $A = 7000 \times e^{0.6}$
Calculate $A$ :
$A \approx 7000 \times 1.822119$ $A \approx 12754.83$
The amount after 10 years with continuous compounding is approximately \$12,754.83.

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