Choose the one alternative that best completes the statement or answers the question.
-A certain country's Consumer Price Index is approximated by $\mathrm { A } ( \mathrm { t } ) = 100 \mathrm { e } ^ { 0.024 \mathrm { t } }$, where $\mathrm { t }$ represents the number of years. Use the function to determine the year in which costs will be $50 \%$ higher than in year 0.
A) year 17
B) year 19
C) year 13
D) year 15

Question

Quizplus · Accepted Answer

The Consumer Price Index is given by $A(t) = 100e^{0.024t}$. To find the year in which costs will be 50% higher than in year 0, we need to solve the equation $A(t) = 150$.Substituting $A(t) = 100e^{0.024t}$, we get $100e^{0.024t} = 150$. Dividing both sides by 100, we get $e^{0.024t} = 1.5$. Taking the natural logarithm of both sides, we get $0.024t = \ln(1.5)$. Solving for $t$, we get $t = \frac{\ln(1.5)}{0.024} \approx 17$.Therefore, the year in which costs will be 50% higher than in year 0 is year 17.

Choose the One Alternative That Best Completes the Statement or Answers