Q. 125 - Precalculus Enhanced with Graphing Utilities

Question

The bacteria in a $4$-liter container double every minute. After $60$ minutes the container is full. How long did it take to fill half the container?

Step-by-Step Solution

Answer

It will take $59minutes$ to fill half the container.

Step 1. Given information.

Consider the given question,

It takes $60$ minutes to fill the full container.

Assume $y=a{e}^{bt}$ to be the formula that considers the growth of bacteria.

As the bacteria doubles every minute, then,

$$\begin{gathered} 2a=a{e}^b \\ b=\ln \left(2\right) \\ b=0.693 \end{gathered}$$

Step 2. Substitute the values in the formula.

Substitute $$\begin{gathered} t=60, \\ b=0.693 \end{gathered}$$ in the formula,

$$\begin{gathered} 4=a{e}^{0.693\times 60} \\ a=3.469\times {10}^{-18} \end{gathered}$$

Assume the container is half filed or it is $2$liters. Then,

$$\begin{gathered} 2=3.469\times {10}^{-18}{e}^{0.693\times t} \\ t=59minutes \end{gathered}$$