Problema Solution
To prevent pests, an orchard can have no more than 4 times as many apple trees as peach trees. Also, the number of apple trees plus 2 times the number of peach trees must not exceed 144. The revenue from a single apple tree is $132 and the revenue from a single peach tree is $161. Determine the number of each type of tree that will maximize revenue. What is the maximum revenue?
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let
a = the number of apple trees, a>0
p = the number of peach trees, p>0
an orchard can have no more than 4 times as many apple trees as peach trees:
a <= 4p
the number of apple trees plus 2 times the number of peach trees must not exceed 144:
a + 2p <= 144
The revenue from a single apple tree is $132 and the revenue from a single peach tree is $161 thus the total revenue will be:
F(a, p) = 132a + 161p
we need to maximize the revenue, that is, find the maximum of F(a, p) = 132a + 161p.
First lets find the corner points of the graph:
a <= 4p
a + 2p <= 144
a>= 0
p >= 0
click here to see the graph:
the corner points are:
(0 , 0), (0, 72) and (96, 24)
F(0, 0) = 132*0 + 161*0 = 0
F(0, 72) = 132*0 + 161*72 = 11,592
F(96, 24) = 132*96 + 161*24 =16,536
the maximum revenue is $16,356.
the orchard should have 96 apple and 24 peach trees to maximize the revenue.