Question

a college student plans to take out a $6,000 loan to cover the cost of purchasing a used car. the loan has a 6% annual interest rate compounded continuously, with no payments due for the first two years. determine the balance due at the end of the first two years. $6,764.98 $5,183.28 $5,637.58 $7,166.65

Explanation:

Step1: Recall the formula for continuous compounding

The formula for continuous compounding is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $t$ is the time in years, and $e$ is the base of the natural logarithm (approximately 2.71828).

Step2: Identify the values of P, r, and t

Given that $P = 6000$, $r = 0.06$ (since 6% = 0.06), and $t = 2$.

Step3: Substitute the values into the formula

Substitute $P = 6000$, $r = 0.06$, and $t = 2$ into the formula $A = Pe^{rt}$:
\[
A = 6000 \times e^{0.06 \times 2}
\]

Step4: Calculate the exponent

First, calculate the exponent: $0.06 \times 2 = 0.12$.

Step5: Calculate $e^{0.12}$

Using a calculator, $e^{0.12} \approx 1.12749685$.

Step6: Calculate the final amount

Multiply 6000 by 1.12749685:
\[
A = 6000 \times 1.12749685 \approx 6764.98
\]

Answer:

$6,764.98$