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a manufacturer can produce tape recorders at a cost of $20 apiece. It is estimated that if the tape recorders are sold for x dollars apiece, consumers will buy 120-x of them each month. Use calculus to determine the proce at which the manufacturer's profit will be the greatest.
I set up the problem like this (please double check this)
R=20x(120-x)
R=2400x - 20x^2
R'(x) = 2400-40x
highest point
0=2400-40x
40x=2400
x=60
Thanks for all your help, sometimes I forget to say Thank You.
If you sell at x per item then your profit is (x-20) per item, your sales are (120-x) and so your total profit is y = (x-20)(120-x). This has a maximum when dy/dx = 0, or at the end points of the range of values of x (don't forget that). The endpoints are x=0, y=-2400 and x=120, y=0. The interesting point if when dy/dx = 140-2x = 0, that is, when x=70; then y=2500. Checking the slope of the graph near this point shows that it is a maximum.
Your analysis maximised not the profit (x-20)(120-x) but the sales revenue x(120-x).
Thank you very much. I don't know how to do it sometimes, but I usually know when I'm doing it wrong. LOL^2