Chapter 9.8, Problem 14P

The populations of two different cities are modeled by the equations shown below. The population (in thousands) is represented by $y$ and the number of years since 1970 is represented by $x$.

What year(s) did the cities have the same population? What was the population of both cities during the year(s) of equal population?

Baskinville: $y=x^2-22x+350$

Cryersport: $y=55x-950$

To determine

To find the years did the city have same population.

Answer to Problem 14P

The population is 1910000 in 2022 years $\left(x=1970+52=2022\right)$ and the population is 425000 in 1995 years $\left(x=1970+25=1995\right)$ .

Explanation of Solution

Given information:

The equation is

  $\begin{array}{l}\text{Baskinville:}y=x^2-22x+350\left[\begin{array}{l}y=\text{number of population in thousand}\\ x=\text{number of years}\end{array}\right]\\ \text{Cryersport:}y=55x-950\end{array}$

Calculation: The equation can be obtained as

  $\begin{array}{l}y=x^2-22x+350\left[\text{equation }1\right]\\ y=55x-950\left[\text{equation }2\right]\end{array}$

Substitute the value of y from equation 2 to equation 1

  $\begin{array}{l}{x}^2-22x+350=55x-950\\ x^2-22x+350-55x+950=0\\ x^2-77x+1300=0\end{array}$

Solve the above equation by quadratic formula,

  $\begin{array}{l}x=\frac{-\left(-77\right)\pm \sqrt{{\left(-77\right)}^2-4\left(1\right)\left(1300\right)}}{2\left(1\right)}\left[\text{quadratic formula}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\right]\\ x=\frac{77\pm \sqrt{5929-5200}}{2}\\ x=\frac{77\pm \sqrt{729}}{2}\\ x=\frac{77\pm 27}{2}\left[\text{since }\sqrt{729}=27\right]\\ x=38.5\pm 13.5\left[\text{divide numerator and denominator by 2}\right]\end{array}$

Therefore,

  $\begin{array}{l}x=\left(38.5+13.5\right)\text{ or }x=\left(38.5-13.5\right)\text{ by zero product property}\\ x=52\text{ or }x=25\end{array}$

Solving for the corresponding y value using equation 2

  $\begin{array}{l}y=55x-950\\ \text{put the value of }x=52\\ y=55\left(52\right)-950\\ y=2860-950\\ y=1910\\ \text{again, put the value of }x=25\\ y=55\left(25\right)-950\\ y=1375-950\\ y=425\end{array}$

Hence,

The population is 1910000 in 2022 years $\left(x=1970+52=2022\right)$ and the population is 425000 in 1995 years $\left(x=1970+25=1995\right)$ .