Question

A rectangular sheet of paper 22cm long and 12 cm broad can be curved to form the lateral surface of a right circular cylinder in two ways. Taking \[\pi = \dfrac{{22}}{7}\], the difference between the volumes of the two cylinders thus formed is
(a).200 c.c.
(b).210 c.c.
(c).250 c.c.
(d).252 c.c.

Answer 1

Hint: Here we will be using the given information on the piece of paper dimensions and applying the formula of volume of a right circular cylinder.

Complete step by step solution:
A rectangular sheet of paper 22cm long and 12 cm broad can be curved to form the lateral surface of a right circular cylinder in two ways. We are to find the difference between the volumes of the two cylinders thus formed. In this case, the piece of paper can be curved to form the lateral surface of a right circular cylinder in two ways.
(i) When rolled along the length,
then height=h=22, and the base of the cylinder has circumference \[2\pi r\]=12.
Thus, the radius of the base is
\[\begin{array}{l}
2\pi r = 12\\
r = \dfrac{6}{\pi }
\end{array}\]
Now, we know that the volume of the right circular cylinder has formula \[V = \pi {r^2}h\].
So, the area of the right circular cone, made by the piece of paper when rolled along the length is
\[V = \pi {\left( {\dfrac{6}{\pi }} \right)^2}\left( {22} \right)\]
When rolled along the breadth,
Then height=h=12, and the base of the cylinder has circumference \[2\pi r\]=22.
Thus, the radius of the base is
 \[\begin{array}{l}
2\pi r = 22\\
r = \dfrac{{11}}{\pi }
\end{array}\]
Now, we know that the volume of the right circular cylinder has formula \[V = \pi {r^2}h\].
So, the area of the right circular cylinder, made by the piece of paper when rolled along the length is
\[V = \pi {\left( {\dfrac{{11}}{\pi }} \right)^2}\left( {12} \right)\]
Thus, the difference of volumes is
 \[\begin{array}{l}
 = \pi {\left( {\dfrac{{11}}{\pi }} \right)^2}\left( {12} \right) - \pi {\left( {\dfrac{6}{\pi }} \right)^2}\left( {22} \right)\\
 = \pi \left[ {{{\left( {\dfrac{{11}}{\pi }} \right)}^2}\left( {12} \right) - {{\left( {\dfrac{6}{\pi }} \right)}^2}\left( {22} \right)} \right]\\
 = \dfrac{{11}}{\pi }\left[ {12 \times 11 - 36 \times 2} \right]\\
 = \dfrac{{11}}{\pi }\left( {60} \right)
\end{array}\]
Putting \[\pi = \dfrac{{22}}{7}\], we get
Difference=210 c.c

Note: In problems like these, it is to be always remembered that all the information given can be translated into concepts which can be solved using mensuration formulas.