SOLUTION: The owner of a sidewalk expresso stand find that the weekly profit for their business is a function of the price they charge per cup. If x equals the price(in dollars) of one cup,

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The owner of a sidewalk expresso stand find that the weekly profit for their business is a function of the price they charge per cup. If x equals the price(in dollars) of one cup,       Log On


   

Question 195688: The owner of a sidewalk expresso stand find that the weekly profit for their business is a function of the price they charge per cup. If x equals the price(in dollars) of one cup, the weekly profit is given P(x)= -2900x^2 + 7250 - 2900.
Approximate the maximum profit and the price per cup that produces that profit.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
P%28x%29=+-2900x%5E2+%2B+7250x+-+2900
When the equation is in the form
ax%5E2+%2B+bx+%2B+c, the max or min
of the function occurs at:
x%5Bmax%5D+=+-b%2F%282a%29 (max in this case)
x%5Bmax%5D+=+-7250+%2F+%282%2A%28-2900%29%29
x%5Bmax%5D+=+7250+%2F+5800
x%5Bmax%5D+=+1.25
At maximum profit, the price per cup is $1.25
And to find the maximum profit:
P%28x%29=+-2900x%5E2+%2B+7250x+-+2900
P%281.25%29=+-2900%2A1.25%5E2+%2B+7250%2A1.25+-+2900
P%281.25%29=+-4531.25+%2B+9062.5+-+2900
P%281.25%29=+-4531.25+%2B+9062.5+-+2900
P%281.25%29+=+1631.25
The maximum profit is $1,631.25
I'll plot to check this:
+graph%28+600%2C+600%2C+-5%2C+3%2C+-200%2C+1800%2C+-2900x%5E2+%2B+7250x+-+2900%29+
Looks like I could be right