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The owner of an orange grove must decide when to pick one variety of oranges. She can sell them for $24 a bushel if she sells them now, with each tree yielding an average of 5 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?

Respuesta :

Answer:

3 weeks

Step-by-step explanation:

let the number of weeks be 'x'

The initial price of the bushel = $24

Decrease in price per bushel = $1.50 per week

after x weeks

price of bushel = $24 - 1.50x

Initial yield of bushel = 5

Increase in yield per week = 0.5

after x weeks

yield of bushel = 5 + 0.5x

now,

Return = Price × Number of bushel

or

R = (24 - 1.50x)(5 + 0.5x)

or

R = 120 + 12x - 7.5x - 0.75x²

or

R = 120 + 4.5x - 0.75x²

now,

for point of maxima, differentiating the return with respect to weeks 'x'

we get

R' = 0 + 4.5 - 2(0.75)x = 0

or

⇒ 1.5x = 4.5

or

⇒ x = 3

Hence,

For maximum returns , the oranges should be picked in 3 weeks

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