# A bacteria culture grows with a constant relative growth rate. The bacteria count was 784 after 2...

## Question:

A bacteria culture grows with a constant relative growth rate. The bacteria count was 784 after 2 hours and 117649 after 6 hours. What is the relative growth rate? What was the initial size of the culture? Find an expression for the number of bacteria after {eq}t {/eq} hours. Find the number of cells after 3 hours. When will the population reach 1,000,000?

## Exponential Growth:

Sometimes there are functions useful to described a natural behavior like this where the number of bacteria is growing across the time. These functions are called Exponential Functions and the behavior is called Exponential Growth.

The behavior is given by: {eq}P_{t}=P_{o}e^{kt} {/eq}

What is the relative growth rate?

We know that: {eq}P_{t}=P_{o}e^{kt} {/eq}

At t=2, {eq}P_{2}=784 {/eq}

At t=6, {eq}P_{6}=117649 {/eq}

Then,

{eq}P_{6}=P_{2}e^{k(6-2)} \\ P_{6}=P_{2}e^{4k} \\ 117649=784e^{4k} \\ e^{4k}=\frac{117649}{784} \\ e^{4k}=150.0625 \\ 4k=\ln(150.0625) \\ k=\frac{\ln(150.0625)}{4} \\ k=1.2527 {/eq}

The relative growth rate is 1.2527

What was the initial size of the culture?

{eq}P_{t}=P_{o}e^{kt} \\ P_{6}=P_{o}e^{1.2527(6)} \\ P_{o}=\frac{P_{6}}{e^{7.5162}} \\ P_{o}=\frac{117649}{e^{7.5162}} \\ P_{o}=64.02 \\ P_{o}\approx64 {/eq}

The initial size of the culture is 64 bacteria.

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

{eq}P_{t}=P_{o}e^{kt} \\ P_{t}=64e^{1.2527t} {/eq}

Find the number of cells after 3 hours.

{eq}P_{t}=64e^{1.2527t} \\ P_{3}=64e^{1.2527(3)} \\ P_{3}=2743.48 \\ P_{3}\approx2743 {/eq}

The number of cells after 3 hours is 2743.

When will the population reach 1,000,000?

{eq}P_{t}=64e^{1.2527t} \\ 1000000=64e^{1.2527t} \\ e^{1.2527t}=\frac{1000000}{64} \\ e^{1.2527t}=15625 \\ 1.2527t=\ln(15625) \\ t=\frac{\ln(15625)}{1.2527} \\ t=7.7086 \\ t\approx8 {/eq}

The population will reach 1000000 in 8 hours. 