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Question

A rectangle of perimeter 36cm swept maximum possible volume, when it is revolved about one of its sides. Then which of the following is/are true

A
Dimensions of the rectangle are 12cm, 6cm
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B
Maximum volume swept is 432π cm3
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C
Dimensions of the rectangle are 10cm, 8cm
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D
Maximum volume swept is 864π cm3
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Solution

The correct option is D Maximum volume swept is 864π cm3
Let l be the length, b be the breath of rectangle and revolved about b
Given 2(l+b)=36
l+b=18(i)
Volume swept,V= Volumeof cylinder with radius l and height b
=πl2b
=πl2(18l)
V=π(18l2l3)
V=π(36l3l2),V′′=π(366l)
For V=0
l=12 (l0) and V′′(12)<0
V is maximum when l=12 and b=6
Vmax=864π cm3

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