Chapter 3.1, Problem 53E

Solution

The year when the population of Columbus, Ohio, and Austin Texas are equal.

The population of Columbus, Ohio, and Austin Texas are equal in $2009$ .

Given information:

The table is shown below:

City	1990 Population	2015 Population
Austin, Texas	465,648	931,830
Columbus, Ohio	632,945	850,106

Concept Used:

Let $a$ and $b$ be real number constants. An exponential function in $x$ is a function that can be written in the form $f\left(x\right)=a{b}^x$ where a is nonzero, $b$ is positive, and $b\ne 1$ . The constant $a$ is the initial value of $f$ (the value at $x=0$ ), and $b$ is the base.

Calculation:

Let $P_1\left(t\right)$ be the population of Austin, Texas $t$ years after July 1, 1990.

  $P_1\left(t\right)=P_0\cdot b^{t_1}$ where $P_0$ is the initial population of Austin, Texas and $b$ is the base.

The initial population of Austin, Texas is $P_0=465,648$ .

The value of $t$ is $2015-1990=25$ .

The population of Austin, Texas after $25$ years is $P_1\left(25\right)=931,830$ .

Substitute $25$ for $t$ , $931,830$ for $P_1\left(25\right)$ and $465,648$ for $P_0$ in $P_1\left(t\right)=P_0\cdot b^t$ .

  $\begin{array}{c}{P}_1\left(25\right)=465,648\cdot b^{25}\\ 931,830=465,648b^{25}\\ b=\sqrt[25]{\frac{931,830}{465,648}}\\ \approx 1.0281374\end{array}$

The population of Austin, Texas $P_1\left(t\right)$ is $465,648\cdot {\left(1.0281374\right)}^t$

Let $P_2\left(t\right)$ be the population of Columbus, Ohio $t$ years after July 1, 1990.

  $P_2\left(t\right)=p_0\cdot b^t$ where $p_0$ is the initial population of Columbus, Ohio and $b$ is the base.

The initial population of Columbus, Ohio is $p_0=632,945$ .

The population of Columbus, Ohio after $25$ years is $P_2\left(25\right)=850,106$ .

Substitute $25$ for $t$ , $850,106$ for $P_2\left(25\right)$ and $632,945$ for $p_0$ in $P_2\left(t\right)=p_0\cdot b^t$ .

  $\begin{array}{c}{P}_2\left(25\right)=632,945\cdot b^{25}\\ 850,106=632945b^{25}\\ b=\sqrt[25]{\frac{850106}{632945}}\\ \approx 1.0118689\end{array}$

The population of Columbus, Ohio $P_2\left(t\right)$ is $632,945\cdot {\left(1.0118689\right)}^t$

Ti-83 Calculator:

• Press the button $Y=$ and then write $Y_1=465,648\cdot \left(1.0281374\right)^(X)$ and $Y_2=632,945\cdot \left(1.0118689\right)^\left(X\right)$ .
• Graph the functions. The intersection must show on the screen.
• Press 2ND and then CALC (with the TRACE button). Select ‘5: intersect’.
• At the ‘First curve?’ prompt, use the up and down arrows to select one function. Press the ENTER button.
• At the ‘Second curve?’ prompt, use the up and down arrows to select the second function. Press the ENTER button. Note, if you are only graphing two equations, they will both be automatically selected during steps 4 and 5.
• At the ‘Guess?’ prompt, use the left and right arrows to move the cursor near the intersection and push the ENTER button. The coordinates of the intersection will be at the bottom of the screen!

  

This population model intersects is about $19.245\approx 19$ year.

So, according to this model, the population of Columbus, Ohio and Austin Texas are equal after $1990+19=2009$ .