question 3\nthis question has two parts. firs...
question 3\nthis question has two parts. first, answer part a. then, answer part b.\npart a\nfinance rawan deposits $13,000 into an account that pays 2.6% annual interest compounded every 6 months.\na. write a function to represent the balance a in the account after t years. round to the nearest thousandth if necessary.\npart b\nb. what will be the balance after 4 years?\nc. what will be the balance after 7.5 years?
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 form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years.
Given $P = 13000$, $r=0.026$ (since $2.6\%=0.026$), and $n = 2$ (compounded every 6 months).
So the function is $A(t)=13000(1 +\frac{0.026}{2})^{2t}=13000(1 + 0.013)^{2t}=13000(1.013)^{2t}$.
## Step2: Calculate balance after 4 years
Substitute $t = 4$ into the function $A(t)$.
$A(4)=13000(1.013)^{2\times4}=13000(1.013)^{8}$.
$(1.013)^{8}\approx1.108347$.
$A(4)=13000\times1.108347\approx14408.511$.
## Step3: Calculate balance after 7.5 years
Substitute $t = 7.5$ into the function $A(t)$.
$A(7.5)=13000(1.013)^{2\times7.5}=13000(1.013)^{15}$.
$(1.013)^{15}\approx1.210904$.
$A(7.5)=13000\times1.210904\approx15741.752$.
# Answer:
a. $A(t)=13000(1.013)^{2t}$
b. $\$14408.511$
c. $\$15741.752$