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A bank features a savings account that has an annual percentage rate of [tex]r=5\%[/tex] with interest compounded quarterly. Raquel deposits [tex]\$10,000[/tex] into the account.

The account balance can be modeled by the exponential formula [tex]A = P\left(1+\frac{r}{n}\right)^{nt}[/tex], where [tex]A[/tex] will be the amount in the bank after a certain number of years, [tex]P[/tex] is the original amount put in the bank, [tex]r[/tex] is the annual percentage rate written as a decimal, [tex]n[/tex] is the number of times each year that the interest is compounded, and [tex]t[/tex] is the time in years.

(A) What values should be used for [tex]P[/tex], [tex]r[/tex], and [tex]n[/tex]?
- [tex]P = \$10,000[/tex]
- [tex]r = 0.05[/tex]
- [tex]n = 4[/tex]

(B) How much money will Raquel have in the account in 8 years?

Answer = [tex]\$\square[/tex]

Round answer to the nearest penny.



Answer :

GinnyAnswer
Let's solve this step by step:

### Part (A)
We need to determine the following values from the problem:

- [tex]\( P \)[/tex]: the initial deposit.
- [tex]\( r \)[/tex]: the annual percentage rate written as a decimal.
- [tex]\( n \)[/tex]: the number of times the interest is compounded per year.

Given in the problem:
- Raquel deposits [tex]\( \$ 10,000 \)[/tex], so [tex]\( P = 10000 \)[/tex].
- The annual percentage rate is [tex]\( 5\% \)[/tex]. As a decimal, [tex]\( r = 0.05 \)[/tex].
- The interest is compounded quarterly, which means 4 times a year. So, [tex]\( n = 4 \)[/tex].

So, we have:
- [tex]\( P = 10000 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( n = 4 \)[/tex]

### Part (B)
We need to calculate how much money Raquel will have in the account in 8 years using the formula [tex]\( A = P \left( 1 + \frac{r}{n} \right)^{nt} \)[/tex].

Substituting the known values:
- [tex]\( P = 10000 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( n = 4 \)[/tex]
- [tex]\( t = 8 \)[/tex] (since we are calculating for 8 years)

Applying these into the formula:
[tex]\[ A = 10000 \left( 1 + \frac{0.05}{4} \right)^{4 \cdot 8} \][/tex]

Now calculating the values step-by-step:
1. [tex]\( \frac{r}{n} = \frac{0.05}{4} = 0.0125 \)[/tex]
2. [tex]\( 1 + \frac{r}{n} = 1 + 0.0125 = 1.0125 \)[/tex]
3. [tex]\( n \cdot t = 4 \cdot 8 = 32 \)[/tex]
4. [tex]\( (1.0125)^{32} \)[/tex]

After computing:

[tex]\[ A = 10000 \times (1.0125)^{32} \approx 14881.305085948254 \][/tex]

Rounding to the nearest penny:

[tex]\[ A \approx 14881.31 \][/tex]

So, the amount of money Raquel will have in the account in 8 years is:

[tex]\[ \boxed{14881.31} \][/tex]

### Summary
(A) The values are:
- [tex]\( P = 10000 \)[/tex]
- [tex]\( r = 0.05 \)[/tex]
- [tex]\( n = 4 \)[/tex]

(B) The amount Raquel will have in the account in 8 years is:
[tex]\[ \boxed{14881.31} \][/tex]

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