Chapter 7.5, Problem 68E

To determine

To calculate: the optimal profit.

Answer to Problem 68E

For crop A optimal number of acres is 60

For crop B optimal number of acres is 90

Optimal profit is $33150

Explanation of Solution

Given information:

Two crops A and B are grown. Sum of land of crop A and crop B is at most 150 acres. Total time for trimming crop A and 2 times of crop B is at most 240 hours. Time for picking 30% of crop A and 10% of crop B is at most 30 hours. The profit is $185 per acre for crop A and $245 per acre for crop B.

Calculation:

Let x be the number of acres for crop A and y be the number of acres for crop B.

Total land is at most 150 acres, so

  $x+y\le 150$

Total time for trimming crop A and 2 times for crop B is at most 240 hours, so

  $x+2y\le 240$

Total time for picking 30% of crop A and 10% of crop B is at most 30 hours, so

  $0.3x+0.1y\le 30$

From above equations we can plot the graph

For $x+y\le 150$ take $x+y=150$ make a table

  $\begin{array}{cccc}x& 0& 150& 100\\ y& 150& 0& 50\end{array}$

Plot the points and join them.

  

The region for $x+y\le 150$ will be the region below the line $x+y=150$ because the point (0,0) is included in the region

For $x+2y\le 240$ take $x+2y=240$ make a table

  $\begin{array}{cccc}x& 0& 240& 100\\ y& 120& 0& 70\end{array}$

Plot the points and join them.

  

The region for $x+2y\le 240$ will be the region below the line $x+2y=240$ because the point (0,0) is included in the region

For $0.3x+0.1y\le 30$ take $0.3x+0.1y=30$ make a table

  $\begin{array}{cccc}x& 0& 100& 10\\ y& 300& 0& 270\end{array}$

Plot the points and join them.

  

The region for $0.3x+0.1y\le 30$ will be the region below the line $0.3x+0.1y=30$ because the point (0,0) is included in the region

  

Profit function (Objective function)

  $P=185x+245y$

Now from graph we have set of 5 points:

  $\left(0,0\right),\left(0,120\right),\left(60,90\right),\left(75,75)\right)\text{and}\left(100,0\right)$

Now put above set of points in profit function to find optimal profit

Put all the points in $P=185x+245y$ and find the optimal value

For $\left(0,0\right),P=185\left(0\right)+245\left(0\right)\Rightarrow P=0$

For $\left(0,120\right),P=185\left(0\right)+245\left(120\right)\Rightarrow P=29400$

For $\left(60.90\right),P=185\left(60\right)+245\left(90\right)\Rightarrow P=33150$

For $\left(75,75\right),P=185\left(75\right)+245\left(75\right)\Rightarrow P=32250$

For $\left(100,0\right),P=185\left(100\right)+245\left(0\right)\Rightarrow P=18500$

Conclusion:

For crop A optimal number of acres is 60

For crop B optimal number of acres is 90

Optimal profit is $33150