Price-Demand equation expressed for revenue as a function of price

Question

This is the Price Demand equation they gave

20p+x=25

 

Heres what I got for the function of price. But I dont think I am right.

R(p)=p(25-20p)

 

It also wants an expression for revenue function for demand

R(x)=20x-x^2

 

Im not totally confident here.

Answer 1

quantity demanded, x, = 25-20p, as p goes up, quantity demanded goes down

or you can solve that equation for p to get

p=(25-x)/20

revenue is price times quantity demanded or

R= px

R(x) = x(25-x)/20 = (25x -x^2)/20 = 5x/4 - x^2/20

or if you want R(p), revenue in terms of price

R=px

R(p)= p(25-20p) = 25p -20p^2

doing it this way allow you to solve for the revenue maximizing price by taking the derivative and set =0

R'(p) = 25 - 40p

40p = 25

p = 25/40 = 5/8 = 0.625