Answer :
We are given an initial investment (the principal) of \[tex]$20,000, an annual interest rate of 5%, and the interest is compounded quarterly. The steps to determine the compound interest function and the balance after 5 years are as follows:
1. First, compute the interest rate per compounding period (quarterly):
$[/tex][tex]$ r = \frac{5\%}{4} = \frac{0.05}{4} = 0.0125. $[/tex][tex]$
2. Next, find the total number of compounding periods over 5 years:
$[/tex][tex]$ n = 4 \times 5 = 20. $[/tex][tex]$
3. The compound interest function in general is given by:
$[/tex][tex]$ A = P \left(1 + r\right)^n, $[/tex][tex]$
where:
- $[/tex]P[tex]$ is the principal,
- $[/tex]r[tex]$ is the interest rate per compounding period, and
- $[/tex]n[tex]$ is the total number of periods.
4. Substituting the values in:
$[/tex][tex]$ A = 20000 \left(1 + 0.0125\right)^{4t} $[/tex][tex]$
Note that with $[/tex]t[tex]$ in years, $[/tex]4t[tex]$ represents the total number of compounding periods. For $[/tex]t = 5[tex]$ years:
$[/tex][tex]$ A = 20000 \left(1.0125\right)^{20}. $[/tex][tex]$
5. Evaluating this expression gives the balance after 5 years:
$[/tex][tex]$ A \approx \$[/tex]25,\!640.74. [tex]$$
Thus, the compound interest function accurately modeling the situation is
$$[/tex] A = 20000 (1.0125)^{4t} [tex]$$[/tex]
and the balance after 5 years is approximately \$25,640.74.
1. First, compute the interest rate per compounding period (quarterly):
$[/tex][tex]$ r = \frac{5\%}{4} = \frac{0.05}{4} = 0.0125. $[/tex][tex]$
2. Next, find the total number of compounding periods over 5 years:
$[/tex][tex]$ n = 4 \times 5 = 20. $[/tex][tex]$
3. The compound interest function in general is given by:
$[/tex][tex]$ A = P \left(1 + r\right)^n, $[/tex][tex]$
where:
- $[/tex]P[tex]$ is the principal,
- $[/tex]r[tex]$ is the interest rate per compounding period, and
- $[/tex]n[tex]$ is the total number of periods.
4. Substituting the values in:
$[/tex][tex]$ A = 20000 \left(1 + 0.0125\right)^{4t} $[/tex][tex]$
Note that with $[/tex]t[tex]$ in years, $[/tex]4t[tex]$ represents the total number of compounding periods. For $[/tex]t = 5[tex]$ years:
$[/tex][tex]$ A = 20000 \left(1.0125\right)^{20}. $[/tex][tex]$
5. Evaluating this expression gives the balance after 5 years:
$[/tex][tex]$ A \approx \$[/tex]25,\!640.74. [tex]$$
Thus, the compound interest function accurately modeling the situation is
$$[/tex] A = 20000 (1.0125)^{4t} [tex]$$[/tex]
and the balance after 5 years is approximately \$25,640.74.
