The initial investment in a savings account in which interest is compounded continuously is $\$ 813$. If the time required to double the amount is $9 \frac { 1 } { 4 }$ years, what is the annual rate? Round your answer to two decimal places.
A) $7.49 \%$
B) $7.70 \%$
C) $13.34 \%$
D) $6.29 \%$
E) $8.89 \%$

Question

Quizplus · Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal amount (initial investment), $r$ is the annual interest rate, and $e$ is the base of the natural logarithm. To double the amount, $A = 2P$, so $2P = Pe^{rt}$. Canceling $P$ from both sides and solving for $r$ gives $r = \frac{\ln(2)}{t}$. Substituting $t = 9.25$ years, $r = \frac{\ln(2)}{9.25} \approx 0.0749$ or $7.49\%$.

The Initial Investment in a Savings Account in Which Interest $813\$ 813$813

The Initial Investment in a Savings Account in Which Interest $\$ 813$