Question

A certain strain of bugs had a population of 42. One week later the population had then risen to 78.A) Develop the function of the form p = ae^kt that represents the population after t weeks. B) use the function to predict when the population will reach 500.

Answer 1

ANSWER:

A)

$p=42\cdot e^{0.619t}$

B) week 4

STEP-BY-STEP EXPLANATION:

We have that the function has the following exponential form:

$p=ae^{kt}$

A)

We must calculate the value of k, replacing all the corresponding values, like this:

$\begin{gathered} 78=42\cdot e^{k\cdot1} \\ e^k=\frac{78}{42} \\ k=\ln \: \mleft(\frac{78}{42}\mright) \\ k=0.619 \end{gathered}$

Therefore, the function would be:

$p=42\cdot e^{0.619t}$

B)

To calculate the time it takes to reach 500, we must plug in the value of p and solve for t, like this:

$\begin{gathered} 500=42\cdot\: e^{0.619t} \\ e^{0.619t}=\frac{500}{42} \\ 0.619t=\ln \: \mleft(\frac{500}{42}\mright) \\ t=\frac{\ln \: \mleft(\frac{500}{42}\mright)}{0.619} \\ t=4 \end{gathered}$

Which means that the population will reach a population of 500 during week 4