SOLUTION: A cable TV company has 900 subscribers, each of whom pays $19 per month. On the basis of a survey, the company believes that for each decrease of $0.25 in the monthly rate, it coul

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Question 764746: A cable TV company has 900 subscribers, each of whom pays $19 per month. On the basis of a survey, the company believes that for each decrease of $0.25 in the monthly rate, it could obtain 50 additional subscribers. At what rate will the maximum revenue be obtained, and how many subscribers will there be at that rate?
Answer by ramkikk66(644) About Me  (Show Source):
You can put this solution on YOUR website!

Currently, cable rate = 19 and subscribers = 900. 
For each 0.25 reduction in rate, subscribers will increase by 50.

Let us say the rate is reduced x times (each reduction = 0.25)
So the reduced rate = 19 - 0.25*x
For each reduction, subscribers go up by 50
So the new number of subscribers = 900 + 50*x

New revenue = %2819+-+0.25%2Ax%29%2A%28900+%2B+50%2Ax%29+=+-12.5%2Ax%5E2+%2B+725%2Ax+%2B+17100

For the revenue to be maximum, the first derivative of the expression has to 
be 0, and the second derivative has to be negative.

Revenue = -12.5%2Ax%5E2+%2B+725%2Ax+%2B+17100

First derivative = -2%2A12.5%2Ax+%2B+725+=+-25%2Ax+%2B+725

Solving for -25*x + 725 = 0, x+=+29

Second derivative = -25 which is negative, confirming that it is a max value.

So, the cable rate for maximum revenue = 19+-+0.25%2A29+=+11.75
At this rate, number of customers = 500+%2B+50%2A29+=+2350

Max revenue will be 11.75%2A2350+=+27612.5

Hope you got it. 

:)