Question

1. you borrow $30,000 at a rate of 7% compounded continuously to pay for a new car. how much will you pay the bank at the end of the 6 year loan?

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 form), $t$ is the time in years, and $e$ is the base of the natural logarithm (approximately 2.71828).

Step2: Identify the given values

We are given:

• Principal amount, $P = \$30,000$
• Annual interest rate, $r = 7\% = 0.07$ (converted to decimal)
• Time, $t = 6$ years

Step3: Substitute the values into the formula

Substitute $P = 30000$, $r = 0.07$, and $t = 6$ into the formula $A = Pe^{rt}$:
\[
A = 30000 \times e^{(0.07 \times 6)}
\]

Step4: Calculate the exponent

First, calculate the exponent $0.07 \times 6 = 0.42$. So the formula becomes:
\[
A = 30000 \times e^{0.42}
\]

Step5: Calculate $e^{0.42}$

Using a calculator, $e^{0.42} \approx 1.52196$.

Step6: Calculate the final amount

Multiply the principal by this value:
\[
A = 30000 \times 1.52196 \approx 45658.8
\]

Answer:

You will pay the bank approximately $\$45,658.80$ at the end of the 6 - year loan.