Choose the one alternative that best completes the statement or answers the question.
-Assume the cost of a gallon of milk is $\$ 3.00$. With continuous compounding, find the time it would take the cost to be 4 times as much (to the nearest tenth of a year), at an annual inflation rate of $6 \%$.
A) $23.1 \mathrm { yr }$
B) $0.1 \mathrm { yr }$
C) $0.0 \mathrm { yr }$
D) $8.5 \mathrm { yr }$

Question

Quizplus · Accepted Answer

To find the time it takes for the cost to quadruple with continuous compounding, we use the formula $A = Pe^{rt}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (as a decimal), and $t$ is the time in years. Here, $A = 4P = 4 \times 3 = 12$, $P = 3$, and $r = 0.06$. Rearranging the formula to solve for $t$, we get $t = \frac{\ln(A/P)}{r}$. Plugging in the values, $t = \frac{\ln(4)}{0.06}$, which approximately equals $23.1$ years.

Choose the One Alternative That Best Completes the Statement or Answers