Items are based on the information below:
A cable network provider in a small town has 500 subscribers and he used to collect Rs. 300 per month from each subscriber. He proposes to increase the monthly charges and it is believed from the past experience that for every increase of Rs. 1, one subscriber will discontinue the service. Based on the above information, answer the following question :
The number of subscribers which gives the maximum revenue is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 400
With an increase of Rs x, the revenue is
\(R = (300 + x)(500 - x)\)
\(= 150000 + 200x - x^2\)
For the maximum, set the derivative to zero:
\( \displaystyle \frac{dR}{dx} = 200 - 2x = 0 \Rightarrow x = 100 \)
\( \displaystyle \frac{d^2R}{dx^2} = -2 \lt 0 \)
So R is maximum at x = 100, and the number of subscribers is
\(500 - 100 = 400\)
Why the other options are incorrect
Option (1) 100 — Incorrect. 100 is the increase in the charge (Rs 100), not the number of subscribers.
Option (2) 200 — Incorrect. 200 subscribers means x = 300, giving R = 600 × 200 = 120000, less than the maximum 160000.
Option (3) 300 — Incorrect. 300 subscribers means x = 200, giving R = 500 × 300 = 150000, less than the maximum 160000.