Question

A rectangular paper 11cm by 8cm can be exactly wrapped to cover the curved surface of a cylinder of height 8cm. The volume of the cylinder is
(a) 66\[c{{m}^{3}}\]
(b) 77\[c{{m}^{3}}\]
(c) 88\[c{{m}^{3}}\]
(d) 121\[c{{m}^{3}}\]

Answer 1

Hint: The area of the rectangular paper is equal to the curved surface area of the cylinder, \[\pi rl\]. Equate the formula and find the radius of the cylinder. Take \[\pi =\dfrac{22}{7}\], now with the known values find the volume of the cylinder, \[\pi {{r}^{2}}h\].

Complete step-by-step answer:
The length of rectangular paper = 11cm.
Breadth of rectangular paper = 8cm.
The height of the cylinder given = 8cm.

It is said that the rectangular paper can wrap up the curved surface area of the cylinder.
Hence, the area of the rectangular paper is equal to the curved surface area (CSA) of the cylinder.

i.e. area of paper = CSA of cylinder.
We know that the area of a rectangle = length \[\times \] breadth.
The CSA of a cylinder is given by the formula, \[2\pi rh\].
\[\therefore \] length \[\times \] breadth = \[2\pi rh\]
\[\begin{align}
  & \Rightarrow 11\times 8=2\pi rh\times 8 \\
 & \therefore r=\dfrac{11\times 8}{2\pi \times 8} \\
\end{align}\]
Put, \[\pi =\dfrac{22}{7}\].
\[r=\dfrac{11}{2\pi }=\dfrac{11\times 7}{2\times 22}=\dfrac{7}{4}\]
Hence we got the radius of the cylinder as \[\dfrac{7}{4}\]cm.
We need to find the volume of the cylinder, which is given by the formula, \[V=\pi {{r}^{2}}h\].
\[\therefore \] Volume of the cylinder = \[\pi {{r}^{2}}h\].
Put, \[r=\dfrac{7}{4},\pi =\dfrac{22}{7}\] and h = 8.
\[\therefore V=\dfrac{22}{7}\times {{\left( \dfrac{7}{4} \right)}^{2}}\times 8=\dfrac{22}{7}\times \dfrac{7}{4}\times \dfrac{7}{4}\times 8\]
\[\therefore V=11\times 7=77c{{m}^{3}}\]
Hence we got the volume of the cylinder as \[77c{{m}^{3}}\].
\[\therefore \] Option (b) is the correct answer.

Note: This is one of the basic questions. The only thing to give concern is that the area of the rectangular paper will be equal to CSA of the cylinder as this paper is wrapped around the cylinder. Also learn the formula for CSA of the cylinder. Here we have used the value of \[\pi =\dfrac{22}{7}\], we can also use 3.14 value. But then, we would get decimal values and then will have to round off to get a matching option.