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A bacteria culture initially contains 100 cells and grows at a rate proportional to its size....

Question:

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 330.

(a) Find an expression for the number of bacteria after t hours.

(b) Find the number of bacteria after 2 hours. (Round your answer to the nearest whole number.)

(c) Find the rate of growth after 2 hours. (Round your answer to the nearest whole number.)

(d) When will the population reach 10,000? (Round your answer to one decimal place.)

Exponential Growth:

To solve this problem, we will be using an exponential model. The model will be using will be of the type given below.

$$P(t)=P_o e^{kt} $$

Our first step will be to complete the model by finding the rate of growth k.

Answer and Explanation:

a) As the bacteria population is growing proportional to its size, we can use the following exponential model to find the population after t hours.

$$N(t)=100e^{kt} $$


Here, N(t) is the population after t hours when the initial population, 100, is growing at a rate of k per hour.

We have to find the value of k to complete the model. We can do this as we know that the population rose to 330 in 1 hour.

$$\begin{align} &N(1)=100e^{k}=330\\ &e^{k}=3.3\\ &k=\ln3.3\\ \therefore &N(t)=100e^{t\ln3.3} \end{align} $$


b) The population after 2 hours will be:

$$\begin{align} N(2)&=100e^{2\ln3.3}\\&=1089 \end{align} $$


c) The rate of growth of N(t) after any time t will be given by its derivative:

$$\begin{align} N'(t)&=\frac{\mathrm{d} }{\mathrm{d} t}\left (100e^{t\ln3.3} \right )\\ &=83.7575e^{t\ln 3.3} \end{align} $$

At t=2, this will be:

{eq}\begin{align} N'(2)&=83.7575e^{2\ln 3.3}\\ &=912.119 \end{align} {/eq}


d) The time after which the population will reach 10000 can be found as follows.

$$\begin{align} &100e^{t\ln3.3}=10000\\ &e^{t\ln3.3}=100\\ &t\ln3.3=\ln 100\\ &t\approx 3.857\,\text{hours} \end{align} $$


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
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