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A bacteria culture starts with 280 bacteria and grows at a rate proportional to its size. After 5...

Question:

A bacteria culture starts with 280 bacteria and grows at a rate proportional to its size. After 5 hours there will be 1,400 bacteria.

a. Express the population after t hours as a function of t.

b. What will be the population after 9 hours?

c. How long will it take for the population to reach 2,830?

Exponential Growth Model:

Exponential growth model shows total population after time t. The population increases with an increasing rate giving a j-curve. This mostly occurs when resources are unlimited, and if resources are limited, it exhibits logistic curve.

Answer and Explanation:

a).

This is an exponential growth model. The population as a function of time is given by equation below:

{eq}\displaystyle P_t=P_oe^{kt}\\\text{This is where:}\\t=\text{Time}\\e=2.718...\\P_o=\text{Population before time t}\\P_t=\text{Population after time t}\\k=\text{Constant} {/eq}

Given that;

{eq}P_t=1400\\P_o=280\\t=5 hrs {/eq}

Let's find the value of k:

{eq}\begin{align*} \displaystyle 1400&=280e^{5k}\\\displaystyle \frac{1400}{280}&=e^{5k}\\5&=e^{5k}\\log\,5&=5k\,log\,e\\\displaystyle 5k&=\frac{log\, 5}{log\, e}\\5k&=1.6094\\k&=0.3219\\\\\therefore P_t&=P_oe^{0.3219t} \end{align*} {/eq}


b).

Population after 9 hours:

{eq}\begin{align*} \displaystyle P_9&=280e^{0.3219\times 9}\\&=5074 \end{align*} {/eq}

The total population after 9 hours will be 5074 bacteria.


c).

{eq}\begin{align*} \displaystyle 2830&=280e^{0.3219t}\\\displaystyle e^{0.3219t}&=10.1071\\0.3219t\,log\,e&=log\, 10.1071\\\displaystyle 0.3219t&=\frac{log\, 10.1071}{log\, e}\\0.3219t&=2.3132\\t&=7.186 hrs \end{align*} {/eq}

It will take about 7 hours and 11 minutes for the bacteria to be 2830.


Learn more about this topic:

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Exponential Growth: Definition & Examples

from High School Algebra I: Help and Review

Chapter 6 / Lesson 10
98K

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