Calculating profit (I've tried several times and I keep get it wrong) Please help?

Question

For a certain company, the cost for producing X items is 45x +300 and the revenue for selling X items is 85x-0.5x^2.

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. 

The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2, 4, 6,  or x + 1, x-1). 

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

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Answer 1

Profit = P = R-C

P = -x^2/2 + 85x - (45x + 300)

P = -x^2/2 +85x-45x -300

P = -x^2/2 + 40x -300

P = 50 = -x^2/2 +40x -300

x^2/2 -40x +350 = 0

x = 40/2(1/2) + or - sqr(40^2 -4(350))

x = 40 + or - sqr(200)

x = 40 + or - 10sqr2

x = about 40+ or -14.14) solutions are: (40-10sqr2, 40+10sqr2)

or with a semi-colon, then (20-10sqr2; 20+10sqr2)

x = about 25.86 or 54.14

x = about each make a profit of 50

solutions are (25.86, 54.14)

or if you really want a semi-colon, ;, which is strange, then (25,86; 54.14)

Maximum P is when P' = 0

P' = -x + 40 = 0

x = 40

P(40) = -(40^2)/2 + 40(40) - 300 = -800+1600 - 300 = 500<2500

A profit of 2500 is Not possible, on the information given.

But the Profit function is profit per some time period, such as maybe 1 month

then you could earn a profit of 2500 for a time period of 5 months, 500x5

You might graph the Profit function. It's a downward opening parabola

with vertex = maximum profit point = V(40, 500)