Polynomial Question

Question

For a certain company, the cost for producing x items is 60x+300 and the revenue for selling x items is 100x−0.5x2.

 

The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit!

 

Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.)

 

Part b: Find two values of x  that will create a profit of $50.

 

 

Part c: Is it possible for the company to make a profit of $2,500?

Answer 1

R=100x -.5x^2

C = 60x + 300

Profit = R-C = -5x^2 +40x -300

Max profit is when R'-C'= 0 = 40-x, x=40

Max profit = -.5(40)^2 +40(40 -300 = 1600 -300- 800 = $500

part c) No, it's not possible to make a profit of $2500. At least not with the given information. But revenue and cost are for some specific time period not given in the problem. $500 is the maximum possible for some unspecified time period. Over 5 times as long, and $2500 is possible.

part b) for x=30, profit = 450

for x=20 profit = 300

for x =10 profit= $50

for x = 70 profit = 2800-300-2450= $50

set R-C = 50 = -.5x^2 +40x -300

-.5x^2 + 40x -350 =0 multiply by -2

x^2 -80x +700 = 0 factor (or if you don't see the factors, complete the square or use the quadratic formula)

(x-70)(x-10) =0

x=10 or 70