Question

find the amount that results from the given investment.
$40 invested at 12% compounded continuously after a period of 3 years

after 3 years, the investment results in $\square.
(round to the nearest cent as needed.)

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

Step2: Identify the values

Here, $P = 40$, $r = 0.12$ (since 12% = 0.12), and $t = 3$.

Step3: Substitute the values into the formula

Substitute $P = 40$, $r = 0.12$, and $t = 3$ into $A = Pe^{rt}$:
$A = 40e^{0.12\times3}$

Step4: Calculate the exponent

First, calculate $0.12\times3 = 0.36$. So, $A = 40e^{0.36}$.

Step5: Calculate the value of $e^{0.36}$

Using a calculator, $e^{0.36}\approx1.433329$.

Step6: Calculate the final amount

Multiply 40 by 1.433329: $A = 40\times1.433329 = 57.33316$.

Step7: Round to the nearest cent

Rounding 57.33316 to the nearest cent gives 57.33.

Answer:

57.33