Expert: Paul Klarreich - 3/4/2008

Question
An orchard contains 53 peach trees with each tree yielding an
average of 51 peaches. For each 3 additional trees planted, the
average yield per tree decreases by 9 peaches. How many trees
should be planted to maximize the total yield of the orchard?

Answer
Questioner:   Melinda
Category:  Advanced Math
Private:  No
 
Subject:  Calculus
Question:  An orchard contains 53 peach trees with each tree yielding an
average of 51 peaches. For each 3 additional trees planted, the
average yield per tree decreases by 9 peaches. How many trees
should be planted to maximize the total yield of the orchard?
............................................
Hi, Melinda,

Identify variables, such as:

p = number of additional trees planted.
y = yield per tree (a function of p)
T = total yield (TO BE MAXIMIZED)

y is a linear function of p, with  y(0) = 51 and a slope of -3.

y = 51 - 3p.

T = (number of trees in orchard) times (yield per tree)

T = (53 + p)(51 - 3p)

Basically, that's it. Multiply out. Differentiate. Set dT/dp = 0. Solve.