A Florida citrus grower estimates that if 150 orange trees are planted, each tree will produce 500 oranges. For each additional tree planted on the same acreage, the number of oranges each tree produces will decrease by 10 oranges. Find the maximum number of oranges the citrus grower can produce. How many trees will the citrus grower need to plant?

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The question would not give a valid answer, the question should be in the form:

A Florida citrus grower estimates that if 30 orange trees are planted, each tree will produce 500 oranges. For each additional tree planted on the same acreage, the number of oranges each tree produces will decrease by 10 oranges. Find the maximum number of oranges the citrus grower can produce. How many trees will the citrus grower need to plant?

Solution:

Let x represent the number of orange trees above 30 to be planted on the acreage and let y be the number of oranges. Since for every additional tree there is a decrease of 10 oranges, hence:

y = (30 + x)(500 - 10x)

y = 15000 - 300x + 500x - 10x²

y = 15000 + 200x - 10x²

The number of oranges is maximum at y' = 0. Hence:

y' = 200 - 20x

200 - 20x = 0

20x = 200

x = 10

Second derivative test = y" = -20 (so this is maximal)

Therefore the number of trees needed = 30 + 10 = 40 trees

Maximum number of oranges = (30 + 10)(500 - 10(10)) = 40(400) = 16000 oranges

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