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.The annual yield per orange tree is fairly constant at 270 pounds per tree when the number of trees per acre is 30 or fewer. For each additional tree over 30, the annual yield per tree for all trees on the acre decreases by 3 pounds due to overcrowding. Find the number of orange trees per acre, x, that maximizes the total yield for an acre, T, in pounds. What is the total maximum yield in pounds of oranges

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Answer:

The answer is below

Step-by-step explanation:

Let the yield per hectare be Y and the number of orange trees per acre, x. For fewer than or 30 trees there is a yield of  270 pounds per tree, therefore:

Y(x) = 270    for x ≤ 30

If it is more than 30 trees the yield decreases by 3 pounds, hence:

Y(x) = 270 - 3(x - 30) = 360 - 3x   for x > 30

The total yield T is given as T(x) = x × Y

T(x) = x × 270 = 270x   for x ≤ 30

T(x) =  x(360 - 3x) = 360x - 3x²   for x > 30

The maximum yield is at T'(x) = 0

T'(x) = 360 - 6x

360 - 6x = 0

6x = 360

x = 360/6

x = 60

T(6) = 360(60) - 3(60²) = 10800 pounds

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