# The population of a colony of rabbits grows exponentially. The colony begins with 15 rabbits; 5...

## Question:

The population of a colony of rabbits grows exponentially. The colony begins with 15 rabbits; 5 years later, there are 360 rabbits.

a. Express the population of the colony of rabbits as a function of time in years.

b. How long does it take for the population of rabbits to reach 1000 rabbits?

## Exponential Growth:

If the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself, we say the quantity grows exponentially as time increase.

The population of living organisms is an example of exponential growth. In such cases, we can express the population as a function of time. It is given by {eq}{\color{Blue} {\displaystyle y=ab^t.}} {/eq}

a) Given that the population grows exponentially.

Thus the population of the rabbits can be expressed as a function of time as follows:

{eq}\displaystyle y=ab^t, {/eq} where {eq}a {/eq} being the initial population.

Given that the initial number of rabbits is {eq}15. {/eq}

Hence {eq}\displaystyle y(t)=15b^t. {/eq}

Next our aim is finding {eq}b. {/eq}

After {eq}5 {/eq} years, population is {eq}360. {/eq}

That is {eq}y(5)=360 \Rightarrow 15b^5=360. {/eq}

{eq}\Rightarrow b^5=24. {/eq}

{eq}\therefore \displaystyle b=(24)^{\frac{1}{5}}\approx 1.89. {/eq}

Hence {eq}y(t)=15(1.89)^t {/eq}

b) We need to find the value of {eq}t {/eq} such that the population is {eq}1000. {/eq}

That is:

{eq}1000=15(1.89)^t \Rightarrow (1.89)^t =66.666. {/eq}

Now take logarith to the base {eq}1.89 {/eq} on both the sides, we get:

{eq}t= \log_{1.89}66.666. {/eq}

Using a calculator, we can find the value of {eq}\log_{1.89}66.666. {/eq} It is approximately {eq}6.6. {/eq}

Hence {eq}t=6.6 {/eq} years. 