Shota invests $\$ 2000$ in a certificate of deposit that earns $2 \%$ in interest each year. $\newline$ Write a function that gives the total value $V(t)$ , in dollars, of the investment $t$ years from now. $\newline$ Do not enter commas in your answer. $\newline$ $V(t)=\square \underset{\overline{+n}}{\square}$

Full solution

Q. Shota invests

\$ 2000

in a certificate of deposit that earns

2 \%

in interest each year.

\newline

Write a function that gives the total value

V(t)

, in dollars, of the investment

t

years from now.

\newline

Do not enter commas in your answer.

\newline

V(t)=\square \underset{\overline{+n}}{\square}

Identify Formula: Identify the formula to calculate the total value of an investment with compound interest. $\newline$ The general formula for compound interest is $V(t) = P(1 + \frac{r}{n})^{nt}$ , where: $\newline$ $P$ = principal amount (initial investment) $\newline$ $r$ = annual interest rate (in decimal form) $\newline$ $n$ = number of times the interest is compounded per year $\newline$ $t$ = number of years $\newline$ Since the problem does not specify the compounding frequency, we assume it is compounded once per year ( $n=1$ ).
Convert Rate to Decimal: Convert the annual interest rate from a percentage to a decimal. $\newline$ The given annual interest rate is $2\%$ , which as a decimal is $0.02$ ( $2\% = \frac{2}{100} = 0.02$ ).
Substitute Values: Substitute the given values into the compound interest formula. $\newline$ Since the interest is compounded once per year, $n = 1$ . The principal $P$ is $(\$2000)$ , the annual interest rate $r$ is $0.02$ , and $t$ is the number of years. The formula becomes: $\newline$ $V(t) = 2000(1 + 0.02/1)^{(1*t)}$
Simplify Formula: Simplify the formula. $\newline$ $V(t) = 2000(1 + 0.02)^t$ $\newline$ $V(t) = 2000(1.02)^t$ $\newline$ This is the function that gives the total value $V(t)$ of the investment $t$ years from now.