A peach orchard owner wants to maximize the amount of peaches produced by her or

Question

A peach orchard owner wants to maximize the amount of peaches produced by her orchard.

She cannot simply plant as many trees as she can, since planting more trees will decrease the amount of fruit that each tree produces (the yield of each tree).

She has found that the per-tree yield can be described by the equation

Y = 800 - 30 x.

Here Y is the yield per tree and x is the number of trees planted per acre.

For example, if there were 10 trees planted per acre, each tree would produce 800 - 30 * 10 = 500 peaches.

Find the number of trees per acre that should be planted in order to produce the maximum crop and the resulting total yield.

Number of trees per acre :  trees per acre

Total yield :  peaches per acre

Explanation / Answer

given Y=800-30x is the yield per tree and x is the number of trees planted per acre

total  amount of peaches produced ,A = yield per tree * number of trees

A = (800-30x)x

A=800x-30x2

maximum yield occurs when dA/dx =0 ,d2A/dx2<0

dA/dx=800-30*2x=0

800-60x=0

=>x=800/60

=>x=40/3

=>x=131/3

d2A/dx2<-60<0

total yield =800(40/3)-30(40/3)2

total yield =48000/9

but we cannot plant 1/3 of tree

so we check yeild for x =13 ,x =14

for x =13

total yield =800(13)-30(13)2=5330

for x =14

total yield =800(14)-30(14)2=5320

to produce the maximum crop

Number of trees per acre : 13 trees per acre

Total yield : 5330 peaches per acre

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.