Chapter 9.6, Problem 42P

The weekly profit of a company is modeled by the function $w=-g^2+120g-28.$

The weekly profit, $w$ is dependent on the number of gizmos, g, sold. If the break-even point is when $w=0,$ how many gizmos must the company sell each week in order to break even?

To determine

To find : How many gizmos must the company sell each week in order to break even.

Answer to Problem 42P

The company must sell 120 gizmos each week in order to break even.

Explanation of Solution

Given information :

The weekly profit of the company is modeled by the function $w=-g^2+120g-28$ .

Calculation :

Since, the break even is when $w=0$

Therefore, set the equation to zero and solve for g:

  $-g^2+120g-28=0$

This is a quadratic equation.

For the quadratic equation $a{x}^2+bx+c=0$ the quadratic formula is

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Now, for the given equation

  $\begin{array}{l}-g^2+120g-28=0\\ g^2-120g+28=0\end{array}$

  $a=1$ , $b=-120$ and $c=28$ .

Therefore,

  $\begin{array}{l}g=\frac{-\left(-120\right)\pm \sqrt{{\left(-120\right)}^2-4\left(1\right)\left(28\right)}}{2\left(1\right)}\\ g=\frac{120\pm \sqrt{14400-112}}{2}\\ g=\frac{120\pm \sqrt{14288}}{2}\\ g=\frac{120+\sqrt{14288}}{2}\text{ and }g=\frac{120-\sqrt{14288}}{2}\\ g=119.77\mathrm{and}g=0.23\end{array}$

Take value and round it to its nearest natural number, therefore,

  $g=120$

Hence,

The company must sell 120 gizmos each week in order to break even.