Chapter 4.4, Problem 67E

To determine

To find: the lesser number of shoes could the company procedure and still make the same profit.

Answer to Problem 67E

The company needs to sell 0.7 million shoes to have profit of $25 millions.

Explanation of Solution

Given information:

The profit $P$ for a shoe manufacturer can be modeled by $P=-21x^3+46x$ , where $x$ is the number of shoes produces. Currently the company produces 1 million shoes and makes a profit of $25 million.

Calculation:

The profit needs to be $25 million, so

  $\begin{array}{c}-21x^3+46x=25\\ -21x^3+46x-25=0\end{array}$

Because the sales of 1 million shoes gives profit of $25 million, so 1 is the solution of equation,

  $-21x^3+46x-25=0$

By the factor theorem $x-1$ is a factor of $-21x^3+46x-25$ .

Using synthetic division,

  $\begin{array}{ccccc}1& -21& 0& 46& -25\\ & & -21& -21& 25\\ & -21& -21& 25& 0\end{array}$

So, have $-21x^3+46x-25=\left(x-1\right)\left(-21x^2-21x+25\right)$ .

Now, solve equation $-21x^2-21x+25=0$ .

  $\begin{array}{c}-21x^2-21x+25=0\\ x=\frac{21\pm \sqrt{{21}^2-4\left(-21\right)\left(25\right)}}{2\left(-21\right)}\\ x=\frac{21\pm \sqrt{2541}}{-42}\\ x\approx -1.7,0.7\end{array}$

Therefore, the company needs to sell 0.7 million shoes to have profit of $25 millions.