Question 1194261: 4 A farmer has available 300 hours of labour per week and 800 tons of fertilizer, and he has a maximum of 26 acres for strawberries and 37 acres for tomatoes. An acre of strawberries requires 10 hours of labour and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labour and 20 tons of fertilizer. The profit from an acre of strawberries is $40,000 and the profit from an acre of tomatoes is $30,000. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis.
c. What would be the effect on the number of acres of strawberries and tomatoes to plant, and the maximum profit if the profit from an acre of strawberries was $50,000 instead of $30,000?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Let x = acres of tomatoes
Let y = acres of strawberries
The constraints are....
x>=0; x<=37 max 37 acres of tomatoes
y>=0; y<=23 max 23 acres of strawberries
3x+10y<=300 max 300 hours of labor
20x+8y<=800 max 800 tons of fertilizer
Use Desmos.com to draw the graph. Graph the OPPOSITE of each inequality (e.g., 3x+10y>300); that way the UNSHADED region will be the feasibility region.
The numbers in the problem require slightly ugly arithmetic to find the corners of the feasibility region; so I won't go any farther with specifics about solving the problem.
Note, however, that it is NOT necessary to find the coordinates of all corners of the feasibility region and evaluate the objective function at each one.
It is easy to determine which corner of the feasibility region will produce that maximum value of the objective function (profit), by comparing the slope of the objective function to the slopes of the constraint boundary lines.
The slopes of the two constraint boundary lines are -3/10 and -5/2.
For the first case where the profits are $40,000 per acre for strawberries and $30,000 per acre for tomatoes, the slope of the objective function is -3/4. Because -3/4 is between -3/10 and -5/2, the maximum profit will be where the two slanted constraint lines intersect.
You can do the ugly arithmetic to find the actual numbers of acres of strawberries and tomatoes.
Note your description for the second case is incorrect....
But the slopes of the constraint boundary lines are so much different that making small changes in the profits per acre for either strawberries or tomatoes would not change the result that the maximum profit is where the two constraint boundary lines intersect.
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