An orange grower finds that she can get an average yield of 40 bushels per tree when she plants 20 trees on an acre. 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 the maximum yield?
solve using quadratic functions please
To find the number of trees per acre that would result in the maximum yield, we need to consider the relationship between the number of trees and the yield per tree.
We know that the initial yield per tree is 40 bushels and it decreases by 1 bushel for each additional tree planted. Let's call the number of trees x and the yield per tree y.
If she plants 20 trees on an acre, the yield per tree would be 40 bushels. So, the equation for the yield per tree, based on the number of trees, would be:
y = 40 - (x - 20)
Now, let's define the total yield per acre. It would be the product of the number of trees and the yield per tree:
yield = x * y
Substituting the value of y from the previous equation, we get:
yield = x * (40 - (x - 20))
Expanding the equation:
yield = 40x - x^2 + 800
To find the maximum yield, we need to find the value of x that maximizes this equation. To do that, we can take the derivative of the equation with respect to x and set it to zero:
d(yield)/dx = 40 - 2x = 0
Solving for x:
40 - 2x = 0
2x = 40
x = 20
Therefore, she should plant 20 trees per acre to maximize the yield.