Solve.
-There are currently 75 million cars in a certain country, increasing exponentially by $3.8 \%$ annually. How many years will it take for this country to have 90 million cars? Round to the nearest year.
A) $3 \mathrm{yr}$
B) $71 \mathrm{yr}$
C) $4 \mathrm{yr}$
D) $5 \mathrm{yr}$

Question

Quizplus · Accepted Answer

To solve this, we use the formula for exponential growth: $P = P_0 \cdot e^{rt}$, where $P$ is the future value, $P_0$ is the initial value, $r$ is the rate of growth, and $t$ is the time in years. However, a more straightforward approach for this problem is using the formula $P = P_0 \cdot (1 + r)^t$, where $r = 3.8\% = 0.038$. We want to find $t$ when $P = 90$ million and $P_0 = 75$ million. So, $90 = 75 \cdot (1 + 0.038)^t$.Dividing both sides by 75 gives: $1.2 = (1.038)^t$.Taking the natural logarithm of both sides: $\ln(1.2) = t \cdot \ln(1.038)$.Solving for $t$ gives: $t = \frac{\ln(1.2)}{\ln(1.038)}$.Using a calculator, $t ≈ 5$ years.

Solve -There Are Currently 75 Million Cars in a Certain 3.8%3.8 \%3.8%

Solve
-There Are Currently 75 Million Cars in a Certain $3.8 \%$