Expert: Alon Mandes - 10/11/2008

Question
Thanks for your help on a previous question, I hope its okay if I ask about another.  My problem on the optimization problems is that I always seem to come up with the wrong equations, therefore the entire problem is thrown off.  

A strawberry farmer will receive $36 per bushel of strawberries during the first week of harvesting. Each week after that, the value will drop $0.75 per bushel. The farmer estimates that there are approximately 123 bushels of strawberries in the fields, and that the crop is increasing at a rate of two bushels per week. When should the farmer harvest the strawberries to maximize their value? How many bushels of strawberries will yield the maximum value? What is the maximum value of the strawberries?

Thanks-

Answer
Ok Gracie, let's make it simple :
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.
From here you can calculate the rest of the question.