How Is the Revenue Function R(x) = 500x(30-x) Derived for the Cable Firm?

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The revenue function R(x) = 500x(30-x) for the cable firm is derived from the relationship between price changes and customer acquisition. The firm currently serves 5000 households at $20 per month, and a $1 decrease in price attracts 500 new customers. The number 30 in the function represents the maximum price decrease possible before losing all customers, calculated from the understanding that a $10 increase from $20 would lead to zero customers. The discussion also briefly touches on a separate question regarding the absence of a hole in the graph of a rational function with a common factor, clarified by the fact that x^2 + 1 is never zero. Understanding these concepts is crucial for analyzing revenue and function behavior in mathematics.
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The problem is this

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

Determine the revenue function R

The back of the book says
R(x)=500x(30-x)

I don't see where the 30 comes from, can anyone tell me?
 
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30 is given information. It is obtained by observing that increasing the monthly fee by 10 dollars from 20 would result in 0 households with the service.
 
ah! Thank you. I feel stupid now. how could i overlook that.
 
Well, since my new problem relates to basic functions, I figured I'll just reuse to this topic.

This isn't exactly homework, but I just have a question.

Why does (x^2+1)(x-1)/(x^2 + 1)(x-2) not have a hole in the graph despite having a common factor?
 
Because x2+ 1 is never 0.
 

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