# An orchard contains 62 peach trees with each tree yielding an average of 50 peaches. For each 3 additional trees planted, the average yield per tree decreases by 12 peaches. How many trees should...

An orchard contains 62 peach trees with each tree yielding an average of 50 peaches. For each 3 additional trees planted, the average yield per tree decreases by 12 peaches. How many trees should be planted to maximize the total yield of orchard?

the number of trees is= ( ) give your answer as a whole number

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First, assume a linear relation between the change in the number of trees and the change in the average yield per tree. y = mx + b is the equation of a line.

Let y = the total number of trees in the orchard

Let x = the number of peaches produced per tree

y = 50 and x = 62

50 = 62m + b

The slope, m, is the rate of change of the number of peaches produced per tree as the number of trees increases, therefore:

m = -12/3 = -4

Now we can solve for the y-intercept, b:

50 = 62(-4) + b

b = 50 + 248 = 298

Therefore our equation relating the number of peaches produced in the orchard as a function of the number of trees is:

y = -4x + 298

Let T = the total number of peaches produced in the orchard

The total number of peaches, T, produced in the orchard will be the product of the number of trees and the number of peaches per tree:

`T=xy`

`T=x(-4x+298)`

`T=-4x^2+298x`

When the derivative of T with respect to x `(dT)/dx` is zero, there is an inflection point (i.e., when the direction of curve reverses), which is therefore the point at which there is a local maximum or minimum on the curve.

`(dT)/dx=-8x+298=0`

`8x=298`

`x=37.25~~37`

As there is only one inflection point, it can be assumed this is a maximum and not a minimum from the context of the question. Therefore, with approximately 37 trees the yield of the orchard is at its maximum

However, were there more than one point one could determine whether each inflection point where a maximum or minimum by taking the second derivative:

`(d^2T)/(dx^2)=-8<0`

Therfore the curve is concave down (as the value is negative), and consequently, the inflection point is indeed a maximum.

Let us assume that there exist a linear relation between the change in the number of trees and the change in the average yield per tree i.e. y = ax + b

Where y ,is the total number of trees in the orchard and x , is the number of peaches produced per tree ,By given condition

y=62,x=50 then

62 = 50 a+ b

The slope a , is the rate of change of the number of peaches produced per tree as the number of trees increases, therefore:

a= -12/3 = -4

Now we can solve for the y-intercept, b:

62 = 50(-4) + b

b = 62 + 200 = 262

Thus our equation reduces as a function of the number of trees and yield is:

y = -4x + 262

Let T = the total number of peaches produced in the orchard ,Thus

T=xy

T=x(-4x+262)

T=-4x^2+262x

So for maximum yields,(dT)/(dx)=0 ,But

`(dT)/(dx)=-8x+262`

Thus

-8x+262=0

x=262/8

x=32.75

`(d^2T)/(dx^2)=-8<0 AA x`

**Thus x=32 will give maximum yield.**