please help me with this precal question!?

Question

Demand Equation, R= XP

the price p and the quantity x sold of a certain product obey the demand equation

x= -5p + 100, 0 less than or equal to (p) less than or equal to 20

(a) express the revenue r as a function of x.
(b) what is the revenue if 15 units are sold?
(c) what quantity x maximizes revenue? what is the maximum revenue?
(d) what price should the company charge to maximize revenue?

I DONT UNDERSTAND THIS STUFF VERY WELL SO PLEASE BE DETAILED.

Answer 1

1) You're told that r = xp, so you have to plug in for p in terms of x. Solve your equation:

x = -5p + 100

p = 20 - x/5

Now,

r = xp = x(20 - x/5) = 20x - (x^2)/5

Which is, obviously, a parabola opening downwards.

b) Take the function in (a) and plug 15 in for x.

c) Now use the function in (a) to find the vertex of the parabola, as that is obviously the highest point. The "x" value for the vertex will be the number of units to maximize the revenue, and the "r" value will be that revenue.

d) Now use your solution for p from part a, plug in the value of x you got in part c, and Bob's your uncle.

Note, please, that I have ignored the restrictions on p stated in the problem. This is probably not going to be an issue, but you should be aware that any value for p you get outside the range of 0 to 20 will be invalid, and you'll have to go back and work with the endpoints of the parabola.

One other thing: you might ask your teacher about that restriction that 0 be "less than or equal" to p. The "equal" part means that this company would be happy to sell their product for a price of 0. Granted, the math won't let that happen, but I still question the assumptions there...

Answer 2

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