Question

An orange grove produces a profit of $ 100 per tree when there are 1300 trees planted. Because of​ overcrowding, the profit per tree​ (for every tree in the​ grove) is reduced by 4 cents per tree for each one additional tree planted. How many trees should be planted in order to maximize the total profit of the orange​ grove? What is the maximum​ profit? How many trees should be planted in order to maximize the total profit of the orange​ grove?

Answer 1

Answer:

Step-by-step explanation:

(x) = (1100 + x) (100 - .05(x-1100))

This is a quadratic, graphs as a parabola that opens downward. A maximum cam be found.

The zeros of the function are

(1100 + x) = 0 ..... or ..... [100 - .05(x-1100)] = 0

x = -1100 is the left x-intercept.

[100 - .05(x-1100)] = 0

100 = .05(x-1100)

2000 = x - 1100

x = 3100 is the right intercept.

Maximization of profits is at the mid point of the zeros (x-intercepts)

(3100 + -1100)/2 = 1000

1100 + 1000 = 2100 trees should be planted to maximize profits.

f(x) = (1100 + 1000) (100 - .05(1000-1100))

f(x) = (2000) (105) = 220,500 is the maximum profit.

I hope this helps!