Question

The profit in manufacturing x refrigerators per day, is given by the profit relation P=-3x^2+240x-800 dollars a) How many refrigerators should be made each day to maximize the total profit? b) what is the maximum profit?

Answer 1

To answer part a), we need to find the vertex of the parabola P. This is because P is a downward facing parabola (because of the -3), so the vertex is the highest point/where the profit is highest. But to find the vertex, we have to find the x-intercepts first so:
\[-3x^{2}+240x-800 = 0\]
If we solve this equation, we get that the x-intercepts occur at x= 3.4852 and x = 76.515 (by the quadratic formula). The vertex lines in the middle of these two intercepts so
vertex x-coord = (3.4852+76.515)/2 = 40.

To find the max profit, we simply plug 40 in for x into our given equation. y = 4000

Thus, the factory should make 40 refrigerators per day to maximize profit and their maximum profit per day is $4000.