Maximizing the yield

Question

An orange grove plants 20 trees per acre and gets a yield of 116 bushels of oranges per tree. For each additional tree planted per acre, the yield decreases by 4 bushels per tree. How many trees should be planted per acre to maximize the yield? What is the total yield?

Answer 1

let x= # of new trees

(20+x)(116-4x)

2320+116x-80x-4x²

2320+36x-4x²

use -b/2a for the equation in the form ax²+bx+c, or -4x²+36x+2320

-36/(2)(-4)=-36/-8=4.5 trees but you can't have half of a tree so you use 4 or 5 

 

20 trees produces 20*116=2320 bushels

21 trees produces 21*112=2352 bushels

22 trees produces 22*108=2376 bushels

23 trees produces 23*104=2392 bushels

24 trees produces 24*100=2400 bushels

25 trees produces 25*96=2400 bushels

26 trees produces 26*92=2392 bushels

notice that 24.5*98=2401 bushels, the maximum, but you can't have half of a tree

plant 4 or 5 trees for a total yield of 2400 bushels

 

 

 

 

Answer 2

Yield = (20 + x)(116 - 4x)

 

Where 20+x is the number of trees planted per acre and (116-4x) is the yield per tree (in bushels).

 

Yield = 20*116 - 80x + 116x - 4x²

 

Yield = -4x² + 36x + 2320

 

To find the maximum yield, take the derivative of the Yield wrt x, set it to zero, solve for x:

 

d(Yield)/dx = -8x + 36

 

0 = -8x + 36

 

8x = 36

 

x = 36/8 = 9/2 = 4.5 trees

 

Since we can't plant half a tree, the orange grove should plant an additional 4 or 5 trees per acre, for a total of 24 or 25 trees per acre.  For 24 trees, the total Yield = (20+4)(116-4*4) = 2400 bushels.  For 25 trees, the total Yield = (20+5)(116-4*5) = 2400 bushels, the same as for 4 additional trees.

 

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You can also solve this problem algebraically (without calculus) by noting that the Yield equation is a quadratic.  Since the coefficient of the x² term is negative (-4), its graph is an inverted parabola, so the vertex of the parabola will be the maximum. The vertex is always located at:

 

x = -b/2a

 

Where a is the coefficient of the x² term and b is the coefficient of the x term.  In this case, a=-4, b= 36 so:

 

x = -36/2(-4) = 36/8 = 4.5