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 $10cm,8cm$
• B. Dimensions of the rectangle are $12cm,6cm$
• C. Maximum volume swept is $432\pi c{m}^3$
• D. Maximum volume swept is $864\pi c{m}^3$

Answer 1

Answer: (B), (D)

Maximum volume swept is $864\pi c{m}^3$

Let $l$ be the length, $b$ be the breath of rectangle and revolved about $b$
Given $2(l+b)=36$
$\Rightarrow l+b=18\cdots \left(i\right)$
Volume swept,$V=$ Volumeof cylinder with radius $l$ and height $b$
$=\pi l^{2}b$
$=\pi l^2(18-l)$
$\therefore V=\pi (18l^2-l^3)$
$\Rightarrow V^{\prime }=\pi (36l-3l^2),V^{\prime \prime }=\pi (36-6l)$
For $V^{\prime }=0$
$\Rightarrow l=12(l\ne 0)$ and $V^{\prime \prime }\left(12\right)<0$
$\therefore V$ is maximum when $l=12$ and $b=6$
$\Rightarrow V_{max}=864\pi c{m}^3$