Question

cole wanted to buy a new motorcycle and put $7,000.00 into an account which earns interest compounded continuously. after 1 year, he withdrew the entire balance of $7,528.00 and bought the motorcycle. what was the interest rate on the account? round your answer to the nearest tenth of a percent.

Explanation:

Step1: Recall continuous compounding formula

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal), and $t$ is the time in years.
We know $P = 7000$, $A = 7528$, and $t = 1$. Substitute these values into the formula:
$7528=7000e^{r\times1}$

Step2: Solve for $r$

First, divide both sides by 7000:
$\frac{7528}{7000}=e^{r}$
Simplify the left - hand side: $\frac{7528}{7000}=1.07542857\approx1.0754 = e^{r}$
Then, take the natural logarithm of both sides:
$\ln(1.0754)=\ln(e^{r})$
Since $\ln(e^{r}) = r$, we have $r=\ln(1.0754)$

Calculate $\ln(1.0754)$: Using a calculator, $\ln(1.0754)\approx0.0727$

Step3: Convert to percentage

To convert $r$ from decimal to percentage, multiply by 100:
$r = 0.0727\times100 = 7.27\%$

Step4: Round to the nearest tenth of a percent

Rounding $7.27\%$ to the nearest tenth of a percent gives $7.3\%$

Answer:

7.3%