A cable television firm presently serves 4,600 households and charges $16 per month. A marketing survey indicates that each decrease of $1 in the monthly charge will result in 460 new customers. Let R(x) denote the total monthly revenue when the monthly charge is x dollars. Find the value of x that results in maximum monthly revenue.

Question

Quizplus · Accepted Answer

The total monthly revenue R(x) can be expressed as R(x) = x(4600 + 460(16 - x)), where x is the monthly charge. To find the maximum revenue, we differentiate R(x) with respect to x and set the derivative equal to zero. Simplifying the expression for R(x) gives a quadratic equation in terms of x. The vertex of this quadratic equation, given by x = -b/(2a) in the general form ax^2 + bx + c, will provide the value of x that maximizes the revenue. Substituting the values from the revenue function into this formula leads to the conclusion that the maximum revenue occurs when x = $13.

A Cable Television Firm Presently Serves 4,600 Households and Charges