Question

a cable television firm presently serves 5000 households and charges $20 per month. A merketing survey indicates that each decrease of $1 in the monthly charge will result in 500 new costumers. Let R(x)denote the total monthly revenue when the monthly charge is x dollars.

Determine the revenue function R

Answer 1

let the number of $1 decreases be n
then the monthly cost = 20-n
number of customers = 5000 + 500n

but you want 20-n = x
so n = 20-x
and 5000 + 500n
= 5000 + 500(20-x)
= 5000 + 10000 - 500x
= 15000-500x

so P(x) = x(15000 - 500x) or 15000 - 500x^2

(this is a very common question, but it is strangely worded.
Usually the question would be,
"What should be monthly charge for a maximum revenue ?" )

Answer 2

why do u multiply everything by x at the end? and what is 15000-500x equal to? the number of costumers?

Answer 3

isn't the revenue = cost per month x number of customers?

didn't I define the "number of customers"
as 5000 + 500n
but x was 20-n which gave me n = 20-x

I then subbed that in 5000 + 500n
to get 15000 - 500x as I showed above step by step.

Answer 4

The above is all correct except for one mistake. x(15,000-500x) is not 15,000-500x^2 but rather 15,000x-500x^2. The x was not properly distributed in the original answer.