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)=−0.002x^2+4.5x−800
. How many patterns must it sell each month to have a maximum profit?
_______________ patterns

What is the maximum profit? $____________________.

Answer 1

Well, since you say algebra and not calculus I suppose we have to do it the hard way. That function is a parabola that opens down (sheds water) and we want its vertex. You can do that by completing the square.
−0.002x^2+4.5x−800 = P
divide everything by -.002 to get 1 as coef of x^2
x^2 - 2250 x + 400,000 = -(1/.002)P
x^2 - 2250 x = -(1/.002)P - 400,000
x^2 - 2250 + (2250/2)^2 = -(1/.002)P - 400,000 + 1,265,625
(x-1125) = -(1/.002)P + 1,265,625
now we know that we need to sell 1125
now we need to write the right side as constant *(P-h)
-(1/.002)(P-.002(1265625) = -(1/.002)(P-2531.25)
so the profit is 2,531.25
check my arithmetic !!!

Answer 2

P = −0.002x^2+4.5x−800
dP/dx = -.004 x + 4.5
max or min when dP/dx = 0
x at max = 4.5/.004 = 1125 whew
P= - 2531.25 + 5062.5 - 800
= 1731.25 oh my, arithmetic problem
OH I forgot the -400,000 when I did it with algebra!
x^2 - 2250 + (2250/2)^2 = -(1/.002)P - 400,000 + 1,265,625
(x-1125) = -(1/.002)P + 865,625
= -1/.002(P- .002(865,635) ) = -1/.002(P- 1731.25)
CARAMBA !!!!

Answer 3

Seems I am better with the easy way :)