Question

a certain loan program offers an interest rate of 9% per year, compounded continuously. assuming no payments are made, how much would be owed after seven years on a loan of $1300? do not round any intermediate computations, and round your answer to the nearest cent.

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.
Here, $P = 1300$, $r = 0.09$ (since 9% = 0.09), and $t = 7$.

Step2: Substitute the values into the formula

Substitute $P = 1300$, $r = 0.09$, and $t = 7$ into $A = Pe^{rt}$:
$A = 1300 \times e^{(0.09 \times 7)}$

Step3: Calculate the exponent

First, calculate the exponent: $0.09 \times 7 = 0.63$. So now we have $A = 1300 \times e^{0.63}$.

Step4: Calculate the value of $e^{0.63}$ and then multiply by 1300

Using a calculator, $e^{0.63} \approx 1.87816664$. Then multiply by 1300: $1300 \times 1.87816664 = 2441.616632$.

Step5: Round to the nearest cent

Rounding 2441.616632 to the nearest cent gives 2441.62.

Answer:

$\$2441.62$