Question

A rectangular metal sheet of area ${2m}^2$ is rolled to form a cylinder of volume $\frac{4}{\pi }{m}^3$. Then the radius of cylinder thus formed is ______ m.

• A. $\frac{4}{\pi }$
• B. $\frac{\pi }{4}$
• C. $\pi$
• D. ${\pi }^2$

Answer 1

The correct option is A $\frac{4}{\pi }$

Let, length of rectangle $=l$

Breadth of rectangle $=h$

After rolling $:$

Height of rectangle $=$ Height of cylinder $=h$ ---- ( 1 )

Length of rectangle $=$ Circumference of circle $=l$ ----- ( 2 )

If $r$ is the radius of the cylinder.

Then, Circumference of circular disk $=2\pi r$

$\Rightarrow$ $2\pi r=l$ [ From ( 2 ) ] ---- ( 3 )

$\Rightarrow$ Area of rectangle $=l\times h=2m^2$ [ Given ]

$\Rightarrow$ Area of rectangle $=2\pi r\times h=2$ [ From ( 3 ) ]

$\Rightarrow$ $2\pi rh=2$

$\Rightarrow$ $\pi rh=1$ ---- ( 4 )

Volume of cylinder $=\pi r^{2}h=\frac{4}{\pi }{m}^3$ [ Given ]

$\Rightarrow$ $=\pi r^{2}h=\frac{4}{\pi }$

$\Rightarrow$ $r\left(\pi rh\right)=\frac{4}{\pi }$

From ( 4 ),

$\Rightarrow$ $r=\frac{4}{\pi }$