Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

Maximizing Profit A manufacturer of skis produces    two types: downhill and cross-country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit? What would the maximum profit be if the time available for manufacturing is increased to 48 hours?

Step-by-Step Solution

Answer

The maximum profit is $\$1920$ when the time available for manufacturing is 48 hours. 

Step 1. Given information

The manufacturer produces two types of skis. Let x- denote the number of downhill skis and y- denotes the number of cross-country skis.
If P denotes the total profit from the skis, then P can be expressed as $P=70x+50y$.
This expression is the objective function. 

We have to maximize the value of P subject to the two constraints x and y. As the number of both types of skis produced is greater than 0 , we get the first two constraints as $x\ge 0 and y\ge 0$.

The manufacturing time allotted for the both types of skis is 40 hours. The time for manufacturing one downhill ski is 2 hours and one cross country ski is 1 hour. So, we get the constraint as  $2x+y\le 40$.

Step 2. Finding the value of profit

The finishing time allotted for the both types of skis is 32 hours. The finishing time for one downhill ski is 1 hour and for one cross country ski is 1 hour. So, we get the constraint as  $x+y\le 32$

The linear programming problem is to maximize $P=70x+50y$ subject to the constraints , $x\ge 0,y\ge 0,2x+y\le 4,$and  $x+y\le 32$

Graph the inequalities we obtained and label the corner points.

The shaded portion of the graph represents the set of feasible points.
Now, we have to find the value of the objective function at each corner point.
List the corner points and the corresponding values of the objective function in a table. 

corner point (x,y)	value of profit  $P=70x+50y$
$(0,0)$	$P=70\left(0\right)+50\left(0\right)=0$
$(20,0)$	$P=70\left(20\right)+50\left(0\right)=1400$
$(8,24)$	$P=70\left(8\right)+50\left(24\right)=1760$
$(0,32)$	$P=70\left(0\right)+50\left(32\right)=1600$

From the above table, we can see that the maximum profit is $\$1760$ and it is achieved with 8 downhill skis and 24 cross country skis.
Therefore, the maximum profit is $\$1760$.

When the time allotted for manufacturing is increased to 48 hours, then the inequality $2x+y\le 40$ changes into $2x+y\le 48$. The rest of the constraints remains the same.
So, the linear programming problem is to maximize $P=70x+50y$ subject to the constraints ,

$x\ge 0,y\ge 0,2x+y\le 48$ and  $x+y\le 32$.

Step 3. Graph the inequalities we obtained and label the corner points.

The shaded portion of the graph represents the set of feasible points. 

Now, we have to find the value of the objective function at each corner point.

Corner points (x,y)	Value of Profit
$(0,0)$	$P=70\left(0\right)+50\left(0\right)=0$
$(24,0)$	$P=70\left(24\right)+50\left(0\right)=1680$
$(16,16)$	$P=70\left(16\right)+50\left(16\right)=1920$
$(0,32)$	$P=70\left(0\right)+50\left(32\right)=1600$Th

From the above table, we can see that the maximum profit achieved is $\$1920$.
Therefore, the maximum profit is $\$1920$ when the time available for manufacturing is 48 hours.