Chapter 9.8, Problem 13P

The weekly profits of two different companies selling similar items that opened for business at the same time are modeled by the equations shown below. The profit is represented by $y$ and the number of weeks the companies have been in business is represented by $x$.

According to the projections, what week(s) did the companies have the same profit? What was the profit of both companies during the week(s) of equal profit?

Company A: $y=x^2-70x+3341$

Company X: $y=50x+65$

To determine

To find the week that the company have same profit

Answer to Problem 13P

The profit is $3965 at 78 week and the profit is $2165 at 42 week.

Explanation of Solution

Given information:

The equation is

  $\begin{array}{l}\text{Company A:}y=x^2-70x+3341\left[\begin{array}{l}y=\text{number of profit}\\ x=\text{number of weeks}\end{array}\right]\\ \text{Company B:}y=50x+65\end{array}$

Calculation: The equation can be obtained as

  $\begin{array}{l}y=x^2-70x+3341\left[\text{equation }1\right]\\ y=50x+65\left[\text{equation }2\right]\end{array}$

Substitute the value of y from equation 2 to equation 1

  $\begin{array}{l}{x}^2-70x+3341=50x+65\\ x^2-70x+3341-50x-65=0\\ x^2-120x+3276=0\end{array}$

Solve the above equation by quadratic formula,

  $\begin{array}{l}x=\frac{-\left(-120\right)\pm \sqrt{{\left(-120\right)}^2-4\left(1\right)\left(3276\right)}}{2\left(1\right)}\left[\text{quadratic formula}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\right]\\ x=\frac{120\pm \sqrt{14400-13104}}{2}\\ x=\frac{120\pm \sqrt{1296}}{2}\\ x=\frac{120\pm 36}{2}\left[\text{since }\sqrt{1296}=36\right]\\ x=60\pm 18\left[\text{divide numerator and denominator by 2}\right]\end{array}$

Therefore,

  $\begin{array}{l}x=\left(60+18\right)\text{ or }x=\left(60-18\right)\text{ by zero product property}\\ x=78\text{ or }x=42\end{array}$

Solving for the corresponding y value using equation 2

  $\begin{array}{l}y=50x+65\\ \text{put the value of }x=78\\ y=50\left(78\right)+65\\ y=3900+65\\ y=3965\\ \text{again, put the value of }x=42\\ y=50\left(42\right)+65\\ y=2100+65\\ y=2165\end{array}$

Hence,

The profit is $3965 at 78 week and the profit is $2165 at 42 week.