Chapter 8, Problem 64SGR

To determine

To find: The time that will take to increase the bacteria from 10 to 675.

Answer to Problem 64SGR

It will take $5.8$ days to increase the bacteria from 10 to 675.

Explanation of Solution

Given:

The equation is $y=a{e}^{kt}$ .

The constant $k$ is $0.728$ .

Calculation:

Substitute $675$ for $y$ , $10$ for $a$ and $0.728$ for $k$ in the formula $y=a{e}^{kt}$ to find the time.

  $\begin{array}{c}675=10e^{0.728t}\\ e^{0.728t}=\frac{675}{10}\\ e^{0.728t}=67.5\end{array}$

Take $\ln$ on both sides of the equation $e^{0.728t}=67.5$ .

  $\begin{array}{c}\ln e^{0.728t}=\ln \left(67.5\right)\\ 0.728t=\ln \left(67.5\right)\\ t=\frac{\ln \left(67.5\right)}{0.728}\\ \approx 5.8\end{array}$

Therefore, it will take $5.8$ days.