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? $____________________.

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 !!!!

To find the number of patterns the company must sell each month to have a maximum profit, we need to determine the value of x that corresponds to the maximum point on the graph of the profit function P(x).

The given profit function is P(x) = -0.002x^2 + 4.5x - 800.

The maximum or minimum point of a quadratic function occurs at its vertex. In this case, we are looking for the maximum point. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where the quadratic function is in the form of ax^2 + bx + c = 0.

For the profit function P(x) = -0.002x^2 + 4.5x - 800, the coefficient of x^2 is a = -0.002 and the coefficient of x is b = 4.5.

Using the formula x = -b / (2a), we can calculate:
x = -(4.5) / (2 * (-0.002))
x = 4.5 / 0.004
x = 1125

Therefore, the company must sell 1125 patterns each month to have a maximum profit.

To find the maximum profit, we substitute the value of x = 1125 into the profit function P(x):
P(1125) = -0.002(1125)^2 + 4.5(1125) - 800

Calculating the expression gives us:
P(1125) = -0.002(1,265,625) + 5062.5 - 800
P(1125) = -2531.25 + 5062.5 - 800
P(1125) = 1731.25

Therefore, the maximum profit is $1731.25.

Seems I am better with the easy way :)

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 !!!