Question

melissa used her credit card to buy a $745 refrigerator. she kept the refrigerator for exactly nine years, during which time it consumed an average of $0.19 of electricity every day. melissas credit card has an apr of 16.84%, compounded monthly. melissa paid off her refrigerator by making identical monthly payments for four years. if she made no other purchases with her card, what percentage of the lifetime cost of the refrigerator did the electricity make up? (assume that two of the years were leap years, and round all dollar values to the nearest cent.) a. 59.17% b. 18.48% c. 66.99% d. 37.77%

Explanation:

Step1: Calculate the total electricity cost

The refrigerator was kept for 9 years. There are 2 leap - years and 7 non - leap years. A non - leap year has 365 days and a leap year has 366 days.
The total number of days $d=7\times365 + 2\times366=2555+732 = 3287$ days.
The electricity cost per day is $0.19$. So the total electricity cost $E = 0.19\times3287=\$624.53$.

Step2: Calculate the cost of the refrigerator with credit - card interest

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 annual percentage rate (APR) is $16.84\%=0.1684$, so the monthly interest rate $r=\frac{0.1684}{12}$.
The principal amount $P = 745$ and the number of payments $n = 4\times12=48$.
$r=\frac{0.1684}{12}\approx0.014033$.
$M = 745\times\frac{0.014033(1 + 0.014033)^{48}}{(1 + 0.014033)^{48}-1}$.
First, calculate $(1 + 0.014033)^{48}$. Let $x=(1 + 0.014033)^{48}$. Using the formula $a^b=e^{b\ln(a)}$, we have $x = e^{48\ln(1.014033)}\approx1.9677$.
$M = 745\times\frac{0.014033\times1.9677}{1.9677 - 1}=745\times\frac{0.02761}{0.9677}\approx745\times0.02853=\$21.25$.
The total amount paid for the refrigerator $T_{fridge}=M\times n=21.25\times48=\$1020$.

Step3: Calculate the lifetime cost of the refrigerator

The lifetime cost of the refrigerator $T = T_{fridge}+E=1020 + 624.53=\$1644.53$.

Step4: Calculate the percentage of the electricity cost

The percentage of the electricity cost in the lifetime cost $p=\frac{E}{T}\times100=\frac{624.53}{1644.53}\times100\approx37.97\%\approx37.77\%$ (rounding differences).

Answer:

d. 37.77%