Question

A bank features a savings account that has an annual percentage rate of
% with interest compounded quarterly. Erick deposits $2,000 into the account.

The account balance can be modeled by the exponential formula

, where
is the future value,
is the present value,
is the annual percentage rate written as a decimal,
is the number of times each year that the interest is compounded, and
is the time in years.

(A) What values should be used for
, r, and
?


2000
,


0.032
,


4


(B) How much money will Erick have in the account in
years?
Answer = $
.
Round answer to the nearest penny.

Answer 1

To complete the calculations for your questions, I'll lay out the information and then proceed with the calculation.

### Part (A): Identify the Values
Using the given information:

- **Present Value (P)**: This is the initial deposit made by Erick, which is $2,000.
- **Annual Percentage Rate (r)**: You mentioned the interest rate is a percentage, but you did not specify what it is. The example you provided shows a value of 0.032, which represents an annual interest rate of 3.2%. I will use this for the calculation unless you provide a different rate.
- **Number of Times Compounded Per Year (n)**: Since the interest is compounded quarterly, this value is 4.

So the values are:
- \( P = 2000 \)
- \( r = 0.032 \)
- \( n = 4 \)

### Part (B): Calculate the Future Value

The future value (A) can be calculated using the formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

For this calculation, you still need to provide the number of years (t). Let's say, for example, that you want to calculate the future value after 5 years (you can adjust accordingly based on your needs).

Using:
- \( P = 2000 \)
- \( r = 0.032 \)
- \( n = 4 \)
- \( t = 5 \)

Substituting these values into the formula:

\[
A = 2000 \left(1 + \frac{0.032}{4}\right)^{4 \times 5}
\]

Calculating further:
1. Calculate \( \frac{0.032}{4} = 0.008 \).
2. Then, the expression inside the parentheses becomes \( 1 + 0.008 = 1.008 \).
3. We calculate the exponent: \( 4 \times 5 = 20 \).
4. Thus, we have:
\[
A = 2000 \times (1.008)^{20}
\]

Now, we calculate \( (1.008)^{20} \):
\[
(1.008)^{20} \approx 1.171659 (rounded to six decimal places)
\]

Finally, we find:
\[
A \approx 2000 \times 1.171659 \approx 2343.32
\]

### Final Result
After 5 years, Erick will have approximately $2343.32 in the account when rounded to the nearest penny.

Please revise the number of years used (5 in this example) to your specific need if different.