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?
y = yield/tree
if n>60,
total crop = y = n (120 - 2n)
y = 120 n - 2 n^2
look for dy/dn = 0
0 = dy/dn = 120 -4n
4 n = 120
n = 30
BUT we said yield per plant drop only applied for n > 60 so plant 60
To determine the optimal number of trees the farmer should plant to produce the maximum yield of fruit, we need to find the point at which the total yield is maximized.
Let's assume the number of trees the farmer plants is 'x' and the expected yield per tree is 'y'. Based on the information provided, we can calculate the expected yield per tree as follows:
y = 120 - 2(x - 60)
Now, let's calculate the total yield by multiplying the expected yield per tree by the number of trees:
Total Yield = x * y
Substituting the value of y:
Total Yield = x * (120 - 2(x - 60))
Now, let's simplify the equation and find the maximum point:
Total Yield = x(120 - 2x + 120)
Total Yield = x(240 - 2x)
Total Yield = 240x - 2x^2
To find the maximum point, we need to take the derivative of the function and set it equal to zero:
d(Total Yield)/dx = 240 - 4x = 0
Solving for 'x':
240 - 4x = 0
4x = 240
x = 60
Therefore, the optimal number of trees the farmer should plant to produce the maximum yield of fruit is 60 trees.
To find the number of trees that would result in the maximum yield of fruit, we need to analyze the relationship between the number of trees and the expected yield.
Let's break down the problem step by step:
1. Calculate the expected yield for different numbers of trees:
- For the first 60 trees, the average yield per tree is 120 kg.
- For each additional tree beyond 60, the expected yield decreases by 2 kg per tree due to overcrowding.
2. Create a function to represent the expected yield as a function of the number of trees.
- Let's call the number of trees "x" and the expected yield "y".
- For the first 60 trees, the expected yield is 120 kg per tree.
- For additional trees beyond 60, the expected yield decreases by 2 kg per tree.
- So the function can be represented as:
y = 120 - 2(x - 60)
3. Find the number of trees that will result in the maximum yield.
- In this case, the maximum yield will occur when the expected yield function reaches its highest point.
- To find this point, we can represent the function as a quadratic equation by rearranging the terms:
y = -2x + 300
4. Find the vertex of the quadratic equation.
- The maximum point of a quadratic equation can be found by using the formula: x = -b / (2a).
- In our equation, a = -2 and b = 0, so the formula becomes: x = 0 / (-4) = 0.
- The vertex of the quadratic equation is x = 0, which means that the function's maximum point occurs when x = 0.
5. Interpret the results.
- Since it is not possible to plant zero trees, we can conclude that the maximum yield will occur when no additional trees are planted beyond the initial 60 trees.
Therefore, the farmer should plant up to 60 trees of the particular type to produce the maximum yield of fruit.