1. ## Math promblem help

An entrepreneur needs some on deciding on a business venture. He has determined that the cost, in dollars, to produce x T-Shirts is C(x)=2x+26,x≥0 and the price-demand function, in dollars per shirt, is p(x)=30−2x,0≤x≤15Help him evaluate the venture by finding the following information:

1. Find the profit functionP(x)
2. Find the number of items which need to be sold in order to maximize
3. Find the maximum profit.
4. Find the price to charge per item in order to maximize profit.
5. Find and interpret break-even points.

2. ## Re: Math promblem help

First when you post an equation you need to specify what each of the letters (variables) represents. You have the "cost function", $C(x)=x+ 26$. Am I correct in guessing that "C" is the cost of making x shirts? You also have the "demand function, \$p(x)= 30- 2x4. Am I correct in guessing that "p" is the price at which people would purchase x shirts?

The first question asks for the "profit function", P. (It is not a good idea to use the same letter, even one capital and the other not, to represent two different quantities.) If you sell x shirts for p dollars each, you will bring in a "revenue" of xp dollars. Your profit is that revenue minus the cost, xp(x)- c(x)=x(30- 2x)- (2x+ 26)= -2x^2+ 30x- 2x- 26= -2x^2+ 28x- 26. That's a quadratic function with a negative coefficient on the square. It's graph is a parabola opening downward. Its maximum value is at the vertex of the parabola. You can find that by completing the square or, if you are taking a Calculus course, setting the derivative equal to 0. (2) asks for the x value at that point and (3) asks for the "y" value. (4) is p(x) for that x. Any "break even points" are where revenue equals cost.