A telephone company in a town has $5000$ subscribers on its list and collects fixed rent charges of $R s . 3, 000$ per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company

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Solution

Given that there are $5000$ subscribers and each subscriber pays $3000$ Rs as rent per year
Let us assume that $x$ Rupees are increased in the rent.
So the total cost received by the company is $C = (3000 + x) (5000 - x)$
$\Rightarrow C = 15000000 + 2000 x - x^{2}$
Differentiate it wrt $x$ , we get $\frac{d C}{d x} = 2000 - 2 x$
By equating it to zero , we get $x = 1000$
So the maximum cost received by the company is $4000 \times 4000 = 16000000$ and it receives it when the rent is increased by $1000$ Rs

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