ariana wanted to buy a new guitar. ariana deposited $100.00 into a new savings account that earns interest…

ariana wanted to buy a new guitar. ariana deposited $100.00 into a new savings account that earns interest compounded continuously. after 1 year, she withdrew the entire balance of $109.00 and bought the guitar. what was the interest rate on the account? round your answer to the nearest tenth of a percent.

ariana wanted to buy a new guitar. ariana deposited $100.00 into a new savings account that earns interest compounded continuously. after 1 year, she withdrew the entire balance of $109.00 and bought the guitar. what was the interest rate on the account? round your answer to the nearest tenth of a percent.

Answer

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, and $t$ is the time in years. Here, $P = 100$, $A = 109$, and $t = 1$. So we have the equation $109=100e^{r\times1}$, which simplifies to $109 = 100e^{r}$.

Step2: Solve for $r$

First, divide both sides of the equation $109 = 100e^{r}$ by 100: $\frac{109}{100}=e^{r}$, or $1.09 = e^{r}$. Then, take the natural logarithm of both sides. Since $\ln(e^{r})=r$, we have $r=\ln(1.09)$.

Step3: Calculate the value of $r$ and convert to percentage

Using a calculator, $\ln(1.09)\approx0.086177$. To convert this to a percentage, multiply by 100: $r = 8.6177%$. Rounding to the nearest tenth of a percent, we get $r\approx8.6%$.

Answer:

$8.6$