A farmer is planting a small orchard. He is advised that if he plants up to 60 trees of a particular type, average harvest from each tree will be about 120 kg, but for each additional tree planted the expected yield will go down by an average of 2 kg per tree due to overcrowding. How many trees should he plant in order to produce the maximum yield of fruit?

Question

A farmer is planting a small orchard. He is advised that if he plants up to 60 trees of a particular type, average harvest from each tree will be about 120 kg, but for each additional tree planted the expected yield will go down by an average of 2 kg per tree due to overcrowding. How many trees should he plant in order to produce the maximum yield of fruit?

Answer 1

Let the number of extra trees be x. Then the total yield y is

y=(60+x)(120-2x)

Now just find where y'=0 to get the maximum yield

Answer 2

To determine the number of trees the farmer should plant in order to produce the maximum yield of fruit, we can analyze the situation step by step.

Let's assume 'x' represents the number of additional trees planted beyond the initial 60 trees.

The average harvest per tree can be expressed as follows:
120 - 2x

To find the maximum yield of fruit, we need to consider the total yield from all the trees. We can calculate this by multiplying the average harvest per tree by the total number of trees:
Total yield = (120 - 2x) * (60 + x)

Now, we need to find the value of 'x' that maximizes the total yield. To do this, we can use a concept from calculus called differentiation.

Step 1: Take the derivative of the total yield equation with respect to 'x':
d(Total yield) / dx = d((120 - 2x) * (60 + x)) / dx

Using the product rule of differentiation, we can expand this expression:
d(Total yield) / dx = (120 - 2x) * (d(60 + x)/dx) + (60 + x) * (d(120 - 2x)/dx)

Simplifying this equation further:
d(Total yield) / dx = (120 - 2x) * 1 + (60 + x) * (-2)

Step 2: Set this derivative equation equal to zero to find the critical points (maximum and minimum points):
(120 - 2x) * 1 + (60 + x) * (-2) = 0

120 - 2x - 120 - 2x = 0

-4x = 0

x = 0

Step 3: Determine whether this critical point is a maximum or minimum. To do this, we need to take the second derivative of the total yield equation and substitute the critical point (x=0) into it:

d^2(Total yield) / dx^2 = d(((120 - 2x) * (60 + x))/dx) / dx

Simplifying and substituting x=0:
d^2(Total yield) / dx^2 = d((120 * 60)/dx) / dx

d^2(Total yield) / dx^2 = 0

Since the second derivative of the total yield equation is zero at x=0, we can conclude that it is a maximum point.

Therefore, the farmer should plant 60 additional trees (x=0) on top of the initial 60 trees to produce the maximum yield of fruit.