Mrs. Math’s research has shown that if 100 apple trees are planted, then
the annual revenue is $90 per tree. If more trees are planted they have less
room to grow and generate fewer apples per tree. As a result, the annual
revenue per tree is reduced by $0.70 for each additional tree planted. How
many trees should be planted to maximize the revenue? Round your
answer appropriately. (Hint: Form a quadratic function)
revenue is #trees * revenue-per-tree
So, if there are x trees more than 100,
r(x) = (100+x)(90-0.70x)
so now just find the vertex of that parabola.
To solve this problem, we need to form a quadratic function to represent the revenue generated based on the number of trees planted.
Let's start by defining some variables:
Let x be the number of additional trees planted (in addition to the initial 100 trees).
Let R(x) be the revenue generated per tree if x additional trees are planted.
We are given that the annual revenue per tree is reduced by $0.70 for each additional tree planted. This means that for x additional trees, the revenue per tree is given by:
R(x) = $90 - $0.70x
To find the total revenue, we need to multiply the revenue per tree by the total number of trees planted. Since we have 100 initial trees and x additional trees, the total number of trees planted is 100 + x.
Therefore, the total revenue, denoted as T(x), is given by:
T(x) = (100 + x) * R(x)
T(x) = (100 + x) * ($90 - $0.70x)
To maximize the revenue, we need to find the value of x that maximizes T(x). One way to do this is by finding the vertex of the quadratic function T(x).
The vertex of a quadratic function in the form ax^2 + bx + c is given by:
x = -b / (2a)
In our case, the quadratic function is T(x) = (100 + x) * ($90 - $0.70x), so a = -0.7 and b = 90 + 0.7(100).
Substituting these values into the vertex formula, we have:
x = -(90 + 0.7(100)) / (2 * (-0.7))
Simplifying the expression, we get:
x = -157 / (-1.4)
x = 112.14
Since the number of trees planted cannot be a decimal value, we round x to the nearest whole number. Therefore, the number of trees to be planted to maximize revenue is 112.
Thus, planting 112 additional trees will result in the maximum annual revenue.