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 maximum revenue generated is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 160000
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:
\( \displaystyle \frac{dR}{dx} = 200 - 2x = 0 \Rightarrow x = 100 \)
\( \displaystyle \frac{d^2R}{dx^2} = -2 \lt 0 \)
At x = 100 the charge is Rs 400 and there are 400 subscribers:
\(R = 400 \times 400 = 160000\)
So the maximum revenue is Rs. 160000 per month.
Why the other options are incorrect
Option (1) Rs. 200000 — Incorrect. This is more than the maximum 160000, so it is never reached.
Option (2) Rs. 180000 — Incorrect. This is more than the maximum 160000, so it is never reached.
Option (4) Rs. 150000 — Incorrect. This is the revenue before any increase (300 × 500), not the maximum.