The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $27 a bushel if she sells them now, with each tree yielding an average of 7 bushels. The yield increases by half a bushel per week for the next 5 weeks, but the price per bushel decreases by $1.50 per bushel each week. In how many weeks should the oranges be picked for maximum return?

The oranges should be picked in 2 weeks for maximum return

Explanation:

We assume that the return of the owner is y ($)

Assume that the number of weeks the oranges should be picked to have maximum return is x (weeks). (x≥0)

If collect now, the price for each bushel is $27

As the price per bushel decrease by $1.50 per bushel each week

=> After x weeks, the price of a bushel decrease: 1.5x ($)

=> The price of 1 bushel after x weeks is: 27 - 1.5x ($)

If collect now, each tree can yield 7 bushels

As the yield increases by half a bushel per week for the next 5 weeks

=> After x weeks with x ≤ 5, each trees would yields: 7 + 0.5x (bushels)

The return = The price of each bushes × The quantity of bushels

=>

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We have: if the equation has the form of with a≠0, its maximum value is:

In the equation , we have: a = -0.75; b = 3; c = 189

=>

To look for the number of weeks, we should find x (0≤x≤5) with which y = 192

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⇔ x = 2

The oranges should be picked in 2 weeks for maximum return