Question

question 7 · 1 point
sung lee invests $4,000 at age 18. he hopes the investment will be worth $12,000 when he turns 30. if the interest
compounds continuously, approximately what rate of growth will he need to achieve his goal? round to the nearest tenth of
a percent.

provide your answer below:

%

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 $P = 4000$, $A = 12000$, and $t = 30 - 18 = 12$ years. Plug these values into the formula:
$12000 = 4000e^{12r}$

Step2: Simplify the equation

Divide both sides by 4000:
$\frac{12000}{4000}=e^{12r}$
$3 = e^{12r}$

Step3: Take the natural logarithm of both sides

$\ln(3)=\ln(e^{12r})$
Since $\ln(e^x)=x$, this simplifies to:
$\ln(3)=12r$

Step4: Solve for r

Divide both sides by 12:
$r=\frac{\ln(3)}{12}$
Calculate $\ln(3)\approx1.0986$, so:
$r\approx\frac{1.0986}{12}\approx0.09155$

Step5: Convert to percentage

Multiply by 100 to get the percentage:
$r\approx0.09155\times100 = 9.155\%$
Round to the nearest tenth of a percent: $9.2\%$

Answer:

9.2%