Chapter 9.6, Problem 41PPE

To determine

To explain:

The quadratic equation and the discriminate to solve the model of the dog-walking service

Answer to Problem 41PPE

The quadratic equation is $-5x^2+20x+700$ . Here, discriminate value is negative then the person cannot earn $\$750$ per week.

Explanation of Solution

Given information:

Number of customers per week: $50$

Charges per walk: $\$14$

Here,

The number $\$1$ decrease: $x$

So, the earning is given by the following formula:

  $E=N\left(ppw\right)$

Where,

Number of customers: $E=50+5x$

Price per walk per customer: $\begin{array}{l}ppw=14-\$1\left(x\right)\\ ppw=14-x\end{array}$

So, the eq. modified as:

  $\begin{array}{l}E=\left(50+5x\right)\left(14-x\right)\\ E=700+20x-5x^2\\ E=-5x^2+20x+700\end{array}$

Put the value of discriminate value $E=\$750$ to get the solution of the model of the dog-walking service:

  $\begin{array}{l}750=-5x^2+20x+700\\ 5x^2-20x+50=0\end{array}$

If the discriminate value is greater than or equal to zero then quadratic equation has a solution and it will results for earning the $E=\$750$ .

So, check for $b^2-4ac$

  $\begin{array}{l}{b}^2-4ac={\left(20\right)}^2-4\left(5\right)\left(50\right)\\ b^2-4ac=-600<0\end{array}$

Hence, discriminate value is negative then the person cannot earn $\$750$ per week.