Question

javier deposited $100.00 into a new savings account that earns interest compounded continuously. after 1 year, the balance in the account was $106.00. 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 $P = 100$, $A = 106$, and $t = 1$. Substitute these values into the formula:
$106 = 100e^{r\times1}$

Step2: Solve for $r$

First, divide both sides by 100:
$\frac{106}{100}=e^{r}$
Simplify $\frac{106}{100}$ to $1.06$:
$1.06 = e^{r}$
To solve for $r$, take the natural logarithm of both sides:
$\ln(1.06)=\ln(e^{r})$
Using the property $\ln(e^{x}) = x$, we get:
$\ln(1.06)=r$
Calculate $\ln(1.06)$ (using a calculator, $\ln(1.06)\approx0.0582689$)

Step3: Convert to percentage

To convert $r$ from decimal to percentage, multiply by 100:
$r\times100\approx0.0582689\times100 = 5.82689\%$
Round to the nearest tenth of a percent: $5.8\%$

Answer:

$5.8$