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A telephone company in the town has 5000...

A telephone company in the town has 5000 subscribers on its list and collects fixed rent charges of Rs. 3000 per year from each person. The company proposes to increase the annual rent and it is believed that for every increase of 1 rupee in the rent, one subscriber will be discontinued. Find what increased annual rental will bring the maximum annual income to the company.

A

900

B

1000

C

1200

D

1500

Text Solution

Verified by Experts

The correct Answer is:
B
Let the annual rental be increased by Rs. x.
Then x subscribers will discontinue.
Now, the new annual rental = Rs. `(3000+x)`
and the number of subscribers `=5000-x`
`:.` the total annual income `=(3000+x)(5000-x)`
Let `f(x)=(3000+x)(5000-x)=15000000+2000x-x^(2)`
`:.f'(x)=("d")/("dx")(15000000+2000x-x^(2))`
`=0+2000xx1-2x=2000-2x`
and `f''(x)=("d")/("dx")(2000-2x)=0-2xx1=-2`
Now, f'(x) = 0 gives `2000-2x=0`
`:.x=1000" and "f''(1000)=-2 lt 0`
`:.` by the second derivative test, f is maximum when x = 1000
Hence, the annual rental should be increased by Rs. 1000 for maximum annual income.
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