Question

The number of bacteria, B left parenthesis h right parenthesis, in a certain population increases according to the following function, where time, h, is measured in hours:

B left parenthesis h right parenthesis equals 1425 e to the power of 0.15 h end exponent

How many hours will it take for the bacteria to reach 3700?

Round your answer to the nearest tenth, and do not round any intermediate computations.

Answer 1

Answer:

About 6.4 hours.

Step-by-step explanation:

We are given the function:
$\displaystyle B(h) = 1425e^{0.15h}$

Which measures the population of bacteria B after h hours.

We want to determine the number of hours it will take for the population to reach 3700 bacteria.

Thus, substitute 3700 for B and solve for h:

$\displaystyle \begin{aligned} (3700) & = 1425e^{0.15h} \\ \\ e^{0.15h} & = \frac{3700}{1425} \end{aligned}$

Take the natural log of both sides. This cancels the e on the left-hand side:
$\displaystyle \begin{aligned} \ln\left(e^{0.15h}\right) & = \ln\frac{3700}{1425} \\ \\ 0.15h & = \ln\frac{3700}{1425} \\ \\ h &= \frac{\ln\dfrac{3700}{1425}}{0.15} \\ \\ & \approx 6.4\text{ hours} \end{aligned}$

In conclusion, it will take about 6.4 hours for the population of the bacteria to reach 3700.