Each orange tree grown in California produces 720 oranges per year if not more than 20 trees are planted per acre. For each additional tree planted per acre, the yield per tree decreases by 15 oranges. How many trees per acre should be planted to obtain the greatest number of oranges?
number of extra trees per acre --- n
number of trees per acre = 20+n
yield per tree = 720 - 15n
number of oranges = N = (20+n)(720-15n)
= 14400 + 420n - 15n^2
N ' = 420 - 30n
= 0 for a max of N
30n = 420
n = 14
for the best yield there should be 20+14 or 34 trees per acre
To determine the number of trees per acre that will yield the greatest number of oranges, we need to find the maximum point of the function that represents the relationship between the number of trees and the number of oranges produced.
Let's denote the number of trees per acre as 'x'. We know that for each additional tree planted, the yield per tree decreases by 15 oranges. So, the yield per tree can be calculated as 720 - 15x.
To find the total number of oranges produced per acre, we multiply the yield per tree by the number of trees: Total Oranges = (720 - 15x) * x.
Now, we need to find the value of 'x' that maximizes the total number of oranges. This can be done by finding the vertex of the parabolic graph since the coefficient of x^2 (which is zero in this case) is negative, indicating a downward-opening parabola.
The x-coordinate of the vertex can be found using the formula: x = -b / 2a, where a is the coefficient of x^2 (which is zero in this case) and b is the coefficient of x (which is -15 in this case). So, x = -(-15) / (2*0) = 0.
As we can see, the value of 'x' is 0, which means the graph is actually a linear function. This indicates that there is no maximum point, and therefore, the number of trees per acre should be as large as possible.
In practical terms, this means that to obtain the greatest number of oranges, there should be no limit on the number of trees planted per acre. However, it's important to consider other factors such as space availability, resources, and maintenance requirements when deciding on the number of trees to be planted in a given area.