Word problem (Maximum Return per tree in pear orchard)

[Linty Fresh]

A commercial pear grower must decide on the optimum time to have fruit picked and sold. If the pears are picked now, they wil bring 30 cents per pound with each tree yielding an average of 60 pounds of salable pears. If the average yield per tree increases 6 pounds per tree per week for the next four weeks, but the price drops 2 cents per pound per week, when should the pears be picked to realize the maximum return per tree? What is the maximum return?

OK, here's what I have:

Let x=number of additional weeks

60+6x=pounds of pears yielded per tree.
30-2x=price of pears (in cents)
Yx)=yield (revenue)

Y(x)=(pounds of pears)*(price)=(60+6x)(30-2x)
=1800-120x+180x-12x^2
=1800+60x-12^2

Y'(x)=60-24x

To find maximum value of this equation:

60-24x=0
x=2.5 additional weeks
Y(2.5)=1,875 cents or $18.75

The thing is that I'm not sure if this is right. The maximum yield is only $0.75 more than if we pick the pears right there (x=0)? Could someone double check my math, please? Thanks so much.

 

[skeeter]

your calculations are correct.

that 75 cents is a maximum difference per tree.

I really don't know much about the size of fruit orchards, but I dobelieve that most of them have more than 1 tree.

 

[Linty Fresh]

Per tree!! Right there in my declaration of variables!

Many thanks!