Question

a population of bacteria is growing according to the exponential model $p = 100e^{0.70t}$, where $p$ is the number of colonies and $t$ is measured in hours. after how many hours will 300 colonies be present? (round answer to the nearest tenth.)
a 0.7
b 1.6
c 5.7
d 7.2

Explanation:

Step1: Set up the equation

We know the model is \( P = 100e^{0.70t} \) and we want to find \( t \) when \( P = 300 \). So we set up the equation:
\( 300 = 100e^{0.70t} \)

Step2: Divide both sides by 100

Divide each side of the equation by 100 to simplify:
\( \frac{300}{100}=e^{0.70t} \)
\( 3 = e^{0.70t} \)

Step3: Take the natural logarithm of both sides

Take the natural log (\(\ln\)) of both sides to solve for the exponent. Recall that \(\ln(e^x)=x\):
\( \ln(3)=\ln(e^{0.70t}) \)
\( \ln(3) = 0.70t \)

Step4: Solve for \( t \)

Divide both sides by \( 0.70 \) to isolate \( t \):
\( t=\frac{\ln(3)}{0.70} \)
Calculate \(\ln(3)\approx1.0986\), then:
\( t=\frac{1.0986}{0.70}\approx1.6 \) (rounded to the nearest tenth)

Answer:

B. 1.6