Question

The number of bacteria in a colony is growing exponentially.

Write an exponential function to model the population y of bacteria x hours after 2 P.M.

Short answer

The exponential function to model the population y of bacteria x hours after 2 P.M. is$y=100{\left(2\sqrt{10}\right)}^x$.

Step-by-step solution

Step 1. Given Information.

Given, the number of bacteria in a colony is growing exponentially as below:

An exponential function to model the population y of bacteria x hours after 2 P.M. is to be determined.

Step 2. Explanation.

An exponential function is of the form $y=a{b}^x$

At 2 P.M. the population of bacteria is 100 i.e., when $x=0,y=100$

At 4 P.M. the population of bacteria is 4000 i.e., when $x=2,y=4000$

Plugging the values to get the values of a and b:

$\begin{array}{l}y=a{b}^x\\ 100=a{b}^0\\ a\left(1\right)=100\\ a=100\end{array}$

$\begin{array}{l}y=a{b}^x\\ 4000=\left(100\right)b^2\\ b^2=\frac{4000}{100}\\ b^2=40\\ b=2\sqrt{10}\end{array}$

So, the equation is $y=100{\left(2\sqrt{10}\right)}^x$

Step 3. Conclusion.

Hence the exponential function to model the population y of bacteria x hours after 2 P.M. is $y=100{\left(2\sqrt{10}\right)}^x$