Analyze Profits

Question

For a certain company the cost for producing x items is 40x+300 and the revenue for selling x items is 80x - 0.5x^2

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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 $2,500$?]]

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If you could, could you break down the steps you took to find the answers so that I can more easily learn how to do it next time

Answer 1

Profit = P = R-C

P= -.5x^2 +80x -4x-300

=-.5x^2 +76x -300

rewrite in vertex form

= -.5(x^2 -76x) -300

complete the square. add and subtract 76^2(.5) = 5776/2 = 2888

=-.5(x^2-138x+76^2) -300+2888

=-.5(x-76)^2 +2588

vertex is (76, 2588) = maximum point.

$2588 is max profit, 76 is the profit maximizing output level

answer to part c is yes, $2500 profit is feasible, but just barely

P=50 = -.5x^2 +76x -300

-.5x^2 +76x -300=50

-.5x^2 +76x=250

-.5(x^2 -132 + 76^2) = 250-76^2/2

-.5(x-76)^2 = -2638

(x-76)^2 = -2638/-.5 = 2638(2) = 5276

x-76 =about +/- 72.64

x = 3.36 or 148.64 are both output levels with close to $50 profit

the profit maximizing output level=76 is the midpoint between output levels with $50 profit

another method is use calculus

take the derivative of the profit function and set =0, solve for x

that's the same as setting Marginal Revenue = Marginal Cost

which gives the profit maximizing output level