Q59. - Algebra 2

Question

Every ten years, the Bureau of the Census counts the number of people living in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, this number had grown to 5.31 million.

Write an exponential function that could be used to model the U.S. population y in millions for 1790 to 1800. Write the equation in terms of x, the number of decades x since 1790.

Step-by-Step Solution

Answer

The exponential function that could be used to model the U.S. population y in millions for 1790 to 1800 in terms of x, the number of decades x since 1790 is $y=3.93{\left(\frac{177}{131}\right)}^x$.

Step 1. Given Information.

Given, every ten years, the Bureau of the Census counts the number of people living in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, this number had grown to 5.31 million.

An exponential function that could be used to model the U.S. population y in millions for 1790 to 1800 in terms of x, the number of decades x since 1790 is to be determined.

Step 2. Explanation .

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

In 1790 the population of U.S. was 3.93 million i.e., when $x=0,y=3.93$

In 1800 the population of U.S. was 5.31 million i.e., when $x=1,y=5.31$

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

$\begin{array}{l}y=a{b}^x\\ 3.93=a{b}^0\\ a\left(1\right)=3.93\\ a=3.93\end{array}$

$\begin{array}{l}y=a{b}^x\\ 5.31=\left(3.93\right)b^1\\ b=\frac{5.31}{3.93}\\ b=\frac{177}{131}\end{array}$

So the equation is $y=3.93{\left(\frac{177}{131}\right)}^x$

Step 3. Conclusion .

Hence the exponential function that could be used to model the U.S. population y in millions for 1790 to 1800 in terms of x, the number of decades x since 1790 is $y=3.93{\left(\frac{177}{131}\right)}^x$