Question

A company finds that it can make a profit of P dollars each month by selling x patterns, according to the formula P ( x ) = − .002 x 2 + 5.5 x − 600 . How many patterns must it sell each month to have a maximum profit? To attain maximum profit they must sell patterns. What is the maximum profit? The max profit is $ .

Answer 1

Answer:

It must sell 1,375 patterns each month in order to attain maximum profit, and that maximum profit is $3,181.25

Explanation:

In order to find the number of patterns that will translate into the maximum profit for the compamy, first we have to consider that the equation given is actually a parabola, and its vertex, when we draw it (Y equals profit and X equals units) proposes the maximum profit possible (if the formula continues to describe the company´s profit).

With that in mind, the general equation of a parabola is as follows.

$y(x)=ax^{2} +bx+c$

in our case

a= - 0.002

b= 5.5

c= -600

And the formula, to find the quantity that will provide the maximum profit (x coordinate of the parabola) is as follows.

$x_{vertex}=\frac{-b}{2a}$

Therefore

$x_{vertex}=\frac{-5.5}{2(-0.002)}  =1375$

Now, if we want to find what is going to be this maximum profit in terms of dollars, we just have to substitute x for 1,375 in the equation given, that is.

$y(1375)=-0.002(1375)^{2} +5.5(1375)-600=3181.25$

Best of luck.