An apple farm yields an average of 39 bushels of apples per tree when 21 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?
27
To find the number of trees that should be planted on an acre in order to get the highest yield, we need to analyze the relationship between the number of trees planted and the yield per tree.
Let's denote the number of trees planted on an acre as "x" and the yield per tree as "y".
We know that the initial yield per tree is 39 bushels and decreases by 1 bushel for each additional tree planted. Therefore, the yield per tree can be represented by the equation:
y = 39 - (x - 21)
To find the total yield per acre, we multiply the yield per tree by the number of trees:
total yield = x * y
Substituting the value of y from the equation above, we get:
total yield = x * (39 - (x - 21))
Expanding the equation, we have:
total yield = 39x - x^2 + 441 - 21x
Simplifying further:
total yield = -x^2 + 18x + 441
To maximize the total yield, we need to find the maximum point of this quadratic equation. We can do this by finding the x-coordinate of the vertex using the formula x = -b / (2a), where a = -1 and b = 18:
x = -18 / (2 * -1)
x = 18 / 2
x = 9
Therefore, the number of trees that should be planted on an acre to get the highest yield is 9.
To find the number of trees that should be planted on an acre in order to obtain the highest yield, we need to analyze the relationship between the number of trees and the yield.
Let's start by defining some variables:
Let T be the number of trees per acre.
Let Y be the yield in bushels per tree.
Based on the information given, we know that when 21 trees are planted on an acre, the average yield per tree is 39 bushels. We can write this as an equation:
21T = 39
Now, we also know that each time 1 more tree is planted per acre, the yield decreases by 1 bushel per tree. This means that for each additional tree, the yield decreases by T bushels. We can write this relationship as another equation:
Y = 39 - T
Now, our goal is to find the value of T that maximizes the yield Y.
To do this, we need to find the derivative of Y with respect to T and set it equal to zero to find the critical points. The critical point will correspond to the maximum yield.
dy/dT = -1
Setting dy/dT = 0, we find that T = 0.
However, T cannot be 0 since we are looking for a positive number of trees. So we ignore this solution.
Therefore, there are no critical points for the yield function. The function is decreasing linearly, so the highest yield will occur when we have the maximum number of trees.
To find this maximum number of trees, we solve the equation:
39 - T = 0
T = 39
Thus, planting 39 trees per acre will result in the highest yield.