Question

if $15,000 is invested at a rate of 4.75% compounded continuously for 25 years, what is the final value of the investment? round the answer to the nearest penny. $49,183.11 $48,790.55 $34,183.11 $32,812.5

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 (approximately 2.71828).

Step2: Identify the values of \( P \), \( r \), and \( t \)

Given:

• Principal amount \( P=\$15,000 \)
• Annual interest rate \( r = 4.75\%=0.0475 \) (converted to decimal by dividing by 100)
• Time \( t = 25 \) years

Step3: Substitute the values into the formula

Substitute \( P = 15000 \), \( r=0.0475 \), and \( t = 25 \) into the formula \( A=Pe^{rt} \):
\[
A=15000\times e^{(0.0475\times25)}
\]

Step4: Calculate the exponent

First, calculate the exponent \( 0.0475\times25 \):
\[
0.0475\times25 = 1.1875
\]

Step5: Calculate \( e^{1.1875} \)

Using a calculator, \( e^{1.1875}\approx3.27887 \) (more precise value can be obtained using a calculator with natural logarithm functions)

Step6: Calculate the final amount \( A \)

Multiply the principal by the value of \( e^{1.1875} \):
\[
A = 15000\times3.27887\approx49183.05
\]
(Note: Due to slight differences in the precision of \( e^{1.1875} \) calculation, the value is close to $49,183.11$ when calculated with more precise decimal places of \( e \))

Answer:

\$49,183.11