Question

A population of bacteria is growing according to the equation p(t)=1000e^.04t. Estimate when the population will exceed 1455.

Answer 1

The given equation is:

$P(t)=1000e^{0.04t}$

It is required to find when the population will exceed 1455.

To do this, substitute the given population and solve the resulting equation for t:

Substitute P(t)=1455 into the equation:

$1455=1000e^{0.04t}$

Solve the equation for t:

$\begin{gathered} \text{ Divide both sides by }1000: \\ \frac{1455}{1000}=\frac{1000}{1000}e^{0.04t} \\ \Rightarrow1.455=e^{0.04t} \\ \text{ Swap the sides of the equation:} \\ \Rightarrow e^{0.04t}=1.455 \end{gathered}$

Take the natural logarithm of both sides:

$\begin{gathered} \Rightarrow0.04t=\ln1.455 \\ \Rightarrow t=\frac{\ln1.455}{0.04}\approx9.38 \end{gathered}$

Hence, the population will exceed 1455 when t is about 9.38.

The population will exceed 1455 when t is about 9.38.