Question 764746
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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 = {{{(19 - 0.25*x)*(900 + 50*x) = -12.5*x^2 + 725*x + 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*x^2 + 725*x + 17100}}}

First derivative = {{{-2*12.5*x + 725 = -25*x + 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*29 = 11.75}}}
At this rate, number of customers = {{{500 + 50*29 = 2350}}}

Max revenue will be {{{11.75*2350 = 27612.5}}}

Hope you got it. 

:)

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