A strawberry farmer will receive $36 per bushel of strawberries during the first

Question

A strawberry farmer will receive $36 per bushel of strawberries during the first week of harvesting. Each week after that, the value will drop $0.90 per bushel. The farmer estimates that there are approximately 128 bushels of strawberries in the fields, and that the crop is increasing at a rate of four bushels per week.  When should the farmer harvest the strawberries to maximize their value? (Assume that "during the first week of harvesting" here means week 1.)

        weeks

How many bushels of strawberries will yield the maximum value?  (Round your answer to the nearest integer.)
     bushels

What is the maximum value of the strawberries?  (Round your answer to two decimal places.)
$

Explanation / Answer

Let's 1st see how much bushels the farmer will have every week :
week 1 : 125
week 2 : 125+2
week 3 : 125+2+2
week 4 : 125+2+2+2
week w : 123+2w

Now,Let's 1st see the price of each bushel every week :
week 1 : 36
week 2 : 36-3/4
week 3 : 36-3/4-3/4
week 4 : 36-3/4-3/4-3/4
week w : 36-(w-1)*(3/4)

Good, now the value as function of weeks will be :
V(w)=[123+2w]*[36-(w-1)*(3/4)]
Now let's derive & set it equal to zero :
v'(w)=[123+2w]'*[36-(w-1)*(3/4)]+[123+2w]*[36-(w-1)*(3/4)]'
v'(w)=2*[36-(w-1)*(3/4)]-(3/4)*[123+2w]
v'(w)=72-(2/3)*(w-1)-92.25-(3/2)w
v'(w)=0
72-2w/3+1.5-92.25-1.5w=0
72-3w-92.25=0
3w=20.25
w=6.75 That means at the 6th week day 5.

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