Question
A rectangular sheet of a paper is rolled in two different ways to form two different cylinders. Find the volume of cylinders in each case, if the sheet measures $44 cm \times 33 cm$.
✓
Answer
There are two cases
Case I When the rectangular sheet is rolled along its length, then length of the sheet forms the circumference of its base and breadth of sheet becomes the height of cylinder.
Let $r cm$ be the radius of the base and $h cm$ be the height.
Then, $h=33 cm$
and circumference of the base $=44 cm$
$\begin{array}{l}\Rightarrow \quad 2 \pi r=44 \\ \Rightarrow \quad 2 \times \frac{22}{7} \times r=44 \Rightarrow r=\frac{44 \times 7}{2 \times 22} \Rightarrow r=7 cm\end{array}$
$\therefore$ Volume of the cylinder $=\pi r^2 h$
$=\frac{22}{7} \times 7 \times 7 \times 33=5082 cm^3$
Case II When the rectangular sheet is rolled along its breadth, then breadth of the sheet forms the circumference of its base and length of the sheet becomes the height of the cylinder.
Now, $h=44 cm$ and circumference $=33 cm$
$\begin{array}{ll}\Rightarrow & 2 \pi r=33 \Rightarrow 2 \times \frac{22}{7} \times r=33 \\ \Rightarrow & r=\frac{33 \times 7}{2 \times 22} \Rightarrow r=\frac{21}{4} cm\end{array}$
$\therefore$ Volume of the cylinder $=\pi r^2 h$
$\begin{array}{l}=\frac{22}{7} \times \frac{21}{4} \times \frac{21}{4} \times 44 \\ =3811.5 cm^3\end{array}$
Case I When the rectangular sheet is rolled along its length, then length of the sheet forms the circumference of its base and breadth of sheet becomes the height of cylinder.
Let $r cm$ be the radius of the base and $h cm$ be the height.
Then, $h=33 cm$
and circumference of the base $=44 cm$
$\begin{array}{l}\Rightarrow \quad 2 \pi r=44 \\ \Rightarrow \quad 2 \times \frac{22}{7} \times r=44 \Rightarrow r=\frac{44 \times 7}{2 \times 22} \Rightarrow r=7 cm\end{array}$
$\therefore$ Volume of the cylinder $=\pi r^2 h$
$=\frac{22}{7} \times 7 \times 7 \times 33=5082 cm^3$
Case II When the rectangular sheet is rolled along its breadth, then breadth of the sheet forms the circumference of its base and length of the sheet becomes the height of the cylinder.
Now, $h=44 cm$ and circumference $=33 cm$
$\begin{array}{ll}\Rightarrow & 2 \pi r=33 \Rightarrow 2 \times \frac{22}{7} \times r=33 \\ \Rightarrow & r=\frac{33 \times 7}{2 \times 22} \Rightarrow r=\frac{21}{4} cm\end{array}$
$\therefore$ Volume of the cylinder $=\pi r^2 h$
$\begin{array}{l}=\frac{22}{7} \times \frac{21}{4} \times \frac{21}{4} \times 44 \\ =3811.5 cm^3\end{array}$
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