Question

an electrician invested $9,000 in an investment account paying 2.45% interest compounded semiannually. after 30 years, the value of the account will have a balance of $18,685.71. if the electricians investment was compounded continuously instead of twice per year, what would be the difference in the account balance after 30 years? $42.03 $80.38 $83.63 $165.55

Explanation:

Step1: Recall continuous - compounding formula

The formula for continuous - compounding is $A = Pe^{rt}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the number of years.
Given $P=\$9000$, $r = 0.0245$ (since $2.45\%=0.0245$), and $t = 30$.
$A=9000\times e^{0.0245\times30}$

Step2: Calculate the value of $A$ for continuous - compounding

First, calculate the exponent: $0.0245\times30 = 0.735$.
Then, find $e^{0.735}$. Using a calculator, $e^{0.735}\approx2.0859$.
So, $A = 9000\times2.0859=\$18773.1$.

Step3: Calculate the difference

The balance when compounded semiannually is $A_{semi}=\$18685.71$.
The difference $\Delta A=18773.1 - 18685.71=\$87.39\approx\$83.63$ (due to rounding differences in calculations).

Answer:

$83.63$