Question 1180597
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. (7 marks)
Let x = no. of acres of strawberries
let y = no. of acres of tomatoes 
:
The acre restraints
x =< 26
y =< 37
the labor equation, arrange both so we can graph it
10x + 3y = 300
3y = -10x + 300
y = (-10/3)x + 100
the amt of fertilizer equation
8x + 20y = 800
20y = -8x + 800
x = = -.4x + 40
:
b. Solve this model by using graphical analysis. (8 marks)
{{{ graph( 300, 200, -50, 100, -50, 100, (-10/3)x+100, -.4x+40, 37) }}}
green is fertilizer equation, red is labor, and blue is the tomato constraint
A vertical line at x=26, should be there for the strawberry constraint, but unable to draw that
intersection occurs at x=21, y=30
21 acres of strawberries yields, 21*40000 = $840000
30 acres of tomatoes  yields,  30 * 30000 = $900000
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The maximum profit with these two crops : $1740000
: 
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?---wasn't it $40000.
:
Using the graph, If you planted the max, 26 acres of strawberries,
you could only plant 13 acres of tomatoes:
 26(50000) + 13(30000) = $169000 profit
however using the same acreage 21 and 30
21(50000) + 30(30000) = 1140000 would be max profit