you want to buy a $33,000 car. the company is...
you want to buy a $33,000 car. the company is offering a 4% interest rate for 36 months (3 years). do not round numbers while working on the problem. only round the final answer. since we are working with money, round to the cent. a) what will your monthly payments be? b) how much interest will you pay over the life of the loan?
Answer
# Explanation:
## Step1: Identify loan - payment formula
The formula for the monthly payment of a loan is $M = P\frac{r(1 + r)^n}{(1 + r)^n-1}$, where $P$ is the principal amount, $r$ is the monthly interest rate, and $n$ is the total number of payments.
The principal $P=\$33000$, the annual interest rate $i = 4\%=0.04$, so the monthly interest rate $r=\frac{0.04}{12}$, and the number of months $n = 36$.
## Step2: Calculate monthly payment
First, calculate $(1 + r)^n=(1+\frac{0.04}{12})^{36}$.
Let $x=\frac{0.04}{12}\approx0.003333$. Then $(1 + x)^{36}=\sum_{k = 0}^{36}\binom{36}{k}x^{k}\approx1.12727$.
$M = 33000\times\frac{\frac{0.04}{12}(1+\frac{0.04}{12})^{36}}{(1+\frac{0.04}{12})^{36}-1}$
$M=33000\times\frac{0.003333\times1.12727}{1.12727 - 1}$
$M=33000\times\frac{0.00375}{0.12727}$
$M=\frac{123.75}{0.12727}\approx972.34$
## Step3: Calculate total amount paid
The total amount paid over 36 months is $M\times n=972.34\times36 = 34984.24$.
## Step4: Calculate total interest paid
The total interest $I$ is the total amount paid minus the principal. So $I=34984.24 - 33000=1984.24$.
# Answer:
a) $\$972.34$
b) $\$1984.24$