Q19. - Algebra 2

Question

During the 19th century, rabbits were brought to Australia. Since the rabbits had no natural enemies on that continent, their population increased rapidly. Suppose there were 65,000 rabbits in Australia in 1865 and 2,500,000 in 1867.

Write an exponential function that could be used to model the rabbit population y in Australia. Write the function in terms of x, the number of years since 1865.

Step-by-Step Solution

Answer

The exponential function that could be used to model the rabbit population y in Australia in terms of x, the number of years since 1865 is $y=65000{\left(\sqrt{\frac{500}{13}}\right)}^x$.

Step 1. Given Information.

Given, during the 19th century, rabbits were brought to Australia. Since the rabbits had no natural enemies on that continent, their population increased rapidly. Suppose there were 65,000 rabbits in Australia in 1865 and 2,500,000 in 1867.

An exponential function that could be used to model the rabbit population y in Australia in terms of x, the number of years since 1865 is to be determined.

Step 2. Explanation .

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

In 1865 there were 65000 rabbits i.e., when $x=0,y=65000$

In 1867 there were 2500000 rabbits i.e., when $x=2,y=2500000$

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

$\begin{array}{l}y=a{b}^x\\ 65000=a{b}^0\\ a\left(1\right)=65000\\ a=65000\end{array}$

$\begin{array}{l}y=a{b}^x\\ 2500000=\left(65000\right)b^2\\ b^2=\frac{2500000}{65000}\\ b^2=\frac{500}{13}\\ b=\sqrt{\frac{500}{13}}\end{array}$

So the equation is $y=65000{\left(\sqrt{\frac{500}{13}}\right)}^x$

Step 3. Conclusion .

Hence, the exponential function that could be used to model the rabbit population y in Australia in terms of x, the number of years since 1865 is $y=65000{\left(\sqrt{\frac{500}{13}}\right)}^x$