use the following formula. a = p((1 + r/n)^nt...
use the following formula. a = p((1 + r/n)^nt - 1)/(r/n) suppose that you drive 30,000 miles per year and gas averages $4 per gallon. complete parts a. and b. below. a. what will you save in annual fuel expenses by owning a hybrid car averaging 30 miles per gallon rather than an suv averaging 9 miles per gallon? $9333 (round to the nearest dollar as needed.) b. if you deposit your monthly fuel savings at the end of each month into an annuity that pays 5.2% compounded monthly, how much will have saved at the end of seven years? $ (round to the nearest dollar as needed.)
Answer
# Explanation:
## Step1: Identify values for annuity formula
$P$ is monthly savings, $r = 0.052$ (annual interest rate), $n = 12$ (compounding periods per year), $t=7$ years. First, find monthly savings. Annual savings is $9333$, so monthly savings $P=\frac{9333}{12}=777.75$.
## Step2: Substitute values into annuity - formula
The annuity - formula is $A=\frac{P\left[\left(1 +\frac{r}{n}\right)^{nt}-1\right]}{\frac{r}{n}}$. Substitute $P = 777.75$, $r = 0.052$, $n = 12$, and $t = 7$ into the formula.
$\frac{r}{n}=\frac{0.052}{12}$, $nt=12\times7 = 84$, and $1+\frac{r}{n}=1+\frac{0.052}{12}$.
$\left(1+\frac{0.052}{12}\right)^{84}\approx1.4277$.
$P\left[\left(1+\frac{r}{n}\right)^{nt}-1\right]=777.75\times(1.4277 - 1)=777.75\times0.4277\approx332.14$.
$A=\frac{332.14}{\frac{0.052}{12}}=\frac{332.14\times12}{0.052}\approx76648$.
# Answer:
$76648$