An orange grower finds that she gets an average yield of 40 bushels per tree when she plants 20 trees on an acre of ground. Each time she adds a tree to an acre, the yield per tree decreases by 1 bushel, due to congestion. How many trees per acre should she plant for maximum yield?
To determine the number of trees per acre that would result in the maximum yield, we need to find the ideal point where the total yield is highest. We can approach this problem by analyzing the relationship between the number of trees and the yield per tree.
Let's break down the given information:
- The grower currently plants 20 trees on an acre.
- With 20 trees, the average yield per tree is 40 bushels.
- Adding another tree reduces the yield per tree by 1 bushel.
Now, let's create a table to help us see the pattern:
Number of Trees | Yield per Tree
--------------- | --------------
20 | 40
21 | 39
22 | 38
23 | 37
... | ...
From the table, we can see that the number of trees increases by 1 each time, while the yield per tree decreases by 1 each time. We can interpret this as a linear relationship:
Yield per Tree = 40 - (Number of Trees - 20)
To find the maximum yield, we need to determine the number of trees that leads to the highest total yield. Since total yield is the product of the number of trees and the yield per tree, we can use the formula:
Total Yield = Number of Trees * Yield per Tree
Let's substitute our yield per tree formula into the total yield formula:
Total Yield = Number of Trees * (40 - (Number of Trees - 20))
Simplifying further:
Total Yield = Number of Trees * (60 - Number of Trees)
To maximize the total yield, we need to find the number of trees that gives us the highest value for the total yield. We can do this by finding the vertex of the quadratic equation. The formula for the x-value of the vertex of a quadratic equation in standard form (ax^2 + bx + c) is:
x = (-b) / (2a)
In our case, the quadratic equation is -Number of Trees^2 + 60Number of Trees. Therefore, the x-value (Number of Trees) of the vertex is:
Number of Trees = (-60) / (2 * -1)
Number of Trees = 30
Accordingly, the grower should plant 30 trees per acre in order to achieve the maximum yield.
To find the number of trees per acre that will give the maximum yield, we can use a quadratic function.
Let's assume x represents the number of trees per acre.
Given that each time a tree is added, the yield per tree decreases by 1 bushel, we can express the average yield per tree as (40 - x).
Therefore, the total yield in bushels would be the product of the number of trees (x) and the average yield per tree (40 - x).
Total Yield (T) = x * (40 - x)
To find the maximum yield, we need to find the value of x that maximizes the quadratic equation T.
We can start by expanding the equation:
T = 40x - x^2
To find the maximum value of T, we need to find the vertex of the parabolic graph represented by the equation.
The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the x-coordinate of the vertex as:
x = -b / (2a)
In our case, a = -1, and b = 40.
Substituting the values into the formula:
x = -40 / (2 * -1)
x = -40 / -2
x = 20
Therefore, the maximum yield can be obtained by planting 20 trees per acre.
Note: It is important to consider practical factors such as tree spacing, access, and other environmental factors when determining the actual number of trees to plant. This calculation provides a mathematical estimate based on the given conditions.