solve the problem. round to the nearest cent....
solve the problem. round to the nearest cent. mrs. stevens wants to have $18,000 in the bank in 3 years. if she deposits $9500 today at 4% compounded quarterly for 3 years, how much additional money will she need to add after three years to her investment to make her balance $18000? a. $8356.79 c. $7295.16 b. $0 d. $10,704.84 please select the best answer from the choices provided
Answer
# Explanation:
## Step1: Identify compound - interest formula
The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. Here, $P = 9500$, $r=0.04$, $n = 4$ (compounded quarterly), and $t = 3$.
## Step2: Calculate the future value of the deposit
Substitute the values into the formula:
\[
\begin{align*}
A&=9500\left(1 +\frac{0.04}{4}\right)^{4\times3}\\
&=9500(1 + 0.01)^{12}\\
&=9500\times(1.01)^{12}
\end{align*}
\]
Using a calculator, $(1.01)^{12}\approx1.12682503$. Then $A = 9500\times1.12682503\approx10704.84$.
## Step3: Calculate the additional amount needed
The amount needed to reach $18000$ is $18000 - 10704.84=7295.16$.
# Answer:
C. $\$7295.16$