Question

A rectangular piece is 24cm long and 22cm wide. A cylinder is formed by rolling the paper along its length. The volume of the cylinder is
[a] 1008 $c{{m}^{3}}$
[b] 462 $c{{m}^{3}}$
[c] 267 $c{{m}^{3}}$
[d] 528 $c{{m}^{3}}$

Answer 1

- Hint: When a rectangle is rolled in, to form a cylinder along its length, then the breadth becomes the circumference of the base, and the length becomes the height of the cylinder. Also, use the volume of the cylinder $=\pi {{r}^{2}}h$ and circumference of the base of the cylinder $=2\pi r$.

Complete step-by-step solution -

Since the cylinder is rolled along the length, we have the circumference of base = breadth of cylinder
Let r be the radius of the cylinder.
Using circumference of the base of the cylinder $=2\pi r$, we get
$2\pi r=24$
Dividing both sides by $2\pi $, we get
$r=\dfrac{24}{2\pi }=\dfrac{12}{\pi }$
Also, the length of the rectangle becomes the height of the cylinder
Hence h = l = 22cm
Now, we know that volume of the cylinder $=\pi {{r}^{2}}h$
Using we get
Volume of cylinder $=\pi {{\left( \dfrac{12}{\pi } \right)}^{2}}\times 22$
Using ${{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}$, we get
Volume of cylinder \[=\pi \dfrac{{{12}^{2}}}{{{\pi }^{2}}}\times 22=\dfrac{144}{\pi }\times 22\]
Using $\pi =\dfrac{22}{7}$, we get
Volume of cylinder \[=\dfrac{144}{22}\times 7\times 22=1008c{{m}^{3}}\]
Hence the volume of the cylinder formed = 1008 $c{{m}^{3}}$
Hence option [a] is correct.

Note: [1] A cylinder formed by rolling a rectangle is a right circular cylinder. Hence the formulae of a right circular cylinder are applied. Note that if instead of a rectangular piece of paper it was a parallelogram, then the above formulae will not be applicable.
[2] The length of a rectangle is usually the larger side of the rectangle, and the breadth is usually the smaller side of the rectangle.