Chapter 3, Problem 27SGR

To determine

To calculate: The number of shoes manufactured to maximize the profit.

Answer to Problem 27SGR

The profit is $480. The company should manufacture 12 outdoor shoes and 16 indoor shoes.

Explanation of Solution

Given information:

Each pair of outdoor shoe require 2 hour in step 1 and 1 hour in step 2 and gives a profit of $20. Each pair of indoor shoe require 1 hour in step 1 and 3 hour in step 2 and gives a profit of $15.

40 hours of labor available for step 1 and 60 hours of labor is available for step 2 process.

Calculation:

Consider the provided information thateach pair of outdoor shoe require 2 hour in step 1 and 1 hour in step 2 and gives a profit of $20. Each pair of indoor shoe require 1 hour in step 1 and 3 hour in step 2 and gives a profit of $15.

40 hours of labor available for step 1 and 60 hours of labor is available for step 2 process.

Let x denote the number of outdoor shoes and y denote the number of indoor shoes.

Since, each pair of outdoor shoe require 2 hour in step 1 and 1 hour in step 2 and gives a profit of $20. Each pair of indoor shoe require 1 hour in step 1 and 3 hour in step 2 and gives a profit of $15.

  $\begin{array}{c}2x+y\le 40\\ x+3y\le 60\end{array}$

And,

  $\begin{array}{c}x\ge 0\\ y\ge 0\end{array}$ .

Plot the inequalities and shade the common region $2x+y\le 40,x+3y\le 60,x\ge 0\text{ and }y\ge 0$

  

The profit function that is to be maximized is $f\left(x,y\right)=20x+15y$ .

Substitute the vertices of the feasible region to find the point at which maximum revenue is there.

Substitute $\left(0,0\right)$ in the function $f\left(x,y\right)=20x+15y$ .

  $f\left(0,0\right)=0$

Substitute $\left(12,16\right)$ in the function $f\left(x,y\right)=20x+15y$ .

  $f\left(12,16\right)=480$

Substitute $\left(20,0\right)$ in the function $f\left(x,y\right)=20x+15y$ .

  $f\left(20,0\right)=400$

Substitute $\left(0,20\right)$ in the function $f\left(x,y\right)=20x+15y$ .

  $f\left(0,20\right)=300$

Since, maximum profit is $480. The company should manufacture 12 outdoor shoes and 16 indoor shoes.