Problema Solution
BUSINESS: Maximum Profit A computer dealer can sell 12 personal computers per week at a price of $2000 each. He estimates that each $400 price decrease will result in three more sales per week. If the computers cost him $1200 each, what price should he charge to maximize his profit? How many will he sell at that price?
Answer provided by our tutors
Let 'x' represent the number of $400 decreases.
Then the number of sold computers will be 12 + 3x, and the new price 1200 - 400x.
The equation describing the profit earned per week from selling 12 + 3x computers at a price of 1200 - 400x is:
p = (12 + 3x)*(2000 - 400x) - (12 + 3x)*1200
p = -1200x^2 - 2400x + 9600
We need to find the maximum of the quadratic function p = -1200x^2 - 2400x + 9600 and the value for x for which the maximum is reached.
x max = -b/2a, where a = -1200, b = - 2400
x max = -(-2400)/(2*(-1200))
x max = 1
1200 - 400*1 = $800 per computer
12 + 3*1 = 15 computers
To maximize the profit he should charge $800 per computer. At that price he will sell 15 computers.