Question

A rectangular sheet of paper 88 cm x 40 cm is rolled along its length and a cylinder is formed.
Find the volume of the cylinder so formed.​

Answer 1

${\large{\pmb{\sf{\underline{\maltese \: \: Understanding \: the \: question...}}}}}$

★ This question says we we have to find out the volume of the cylinder so formed from a rectangular sheet of paper whose dimensions are 80 cm and 40 cm as length and breadth respectively. That rectangular sheet of paper is rolled along it's length and a cylinder is formed. As it's rolled means the length be circumference and breadth be the height of the cylinder.

Diagram related to the rectangular sheet of paper-

$\begin{gathered} \sf 80 \: cm \: \: \: \: \: \: \: \: \: \: \: \\ \begin{gathered}\begin{gathered}\boxed{\begin{array}{}\bf { \red{}}\\{\qquad \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{}\\ { \sf{ }}\\ { \sf{ }} \\ \\ { \sf{ }}\end{array}}\end{gathered}\end{gathered} \sf 40 \: cm \end{gathered}$

Diagram related to the rolled form of paper from where cylinder is formed-

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\thicklines\multiput(-0.5,-1)(26,0){2}{\line(0,1){40}}\multiput(12.5,-1)(0,3.2){13}{\line(0,1){1.6}}\multiput(12.5,-1)(0,40){2}{\multiput(0,0)(2,0){7}{\line(1,0){1}}}\multiput(0,0)(0,40){2}{\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\multiput(18,2)(0,32){2}{\sf{r}}\put(9,17.5){\sf{h}}\end{picture}$

${\large{\pmb{\sf{\underline{\maltese \: \: Some \: formulas...}}}}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Volume \: of \: cylinder \: = \: \pi r^{2}h}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Surface \: area \: of \: cylinder \: = \: 2 \pi rh + 2 \pi r^{2}}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Lateral \: area \: of \: cylinder \: = \: 2 \pi rh}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Base \: area \: of \: cylinder \: = \: \pi r^{2}}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Height \: of \: cylinder \: = \: \dfrac{v}{\pi r^{2}}}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Radius \: of \: cylinder \: = \:\sqrt \dfrac{v}{\pi h}}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Perimeter \: of \: rectangle \: = 2(l+b)}}$

$\; \; \; \; \; \; \;{\sf{\leadsto Area \: of \: rectangle \: = L \times B}}$

${\large{\pmb{\sf{\underline{\maltese \: \: Given \: that...}}}}}$

★ Length of rectangular sheet of paper = 80 centimetres.

★ Breadth of rectangular sheet of paper = 40 centimetres.

★ Rectangular sheet of paper is rolled along it's length and a cylinder is formed.

${\large{\pmb{\sf{\underline{\maltese \: \: To \: find...}}}}}$

★ The volume of the cylinder so formed from a rectangular sheet of paper

${\large{\pmb{\sf{\underline{\maltese \: \: Solution...}}}}}$

★ The volume of the cylinder so formed from a rectangular sheet of paper = 24617.6 centimetres cube

${\large{\pmb{\sf{\underline{\maltese \: \: Using \; concepts...}}}}}$

${\small{\underline{\boxed{\sf{Circumference \: of \: circle \: = 2 \pi r}}}}}$

${\small{\underline{\boxed{\sf{Volume \: of \: cylinder \: = \pi r^{2} h}}}}}$

${\large{\pmb{\sf{\underline{\maltese \: \: Full \; Solution...}}}}}$

~ As we already know that the when paper rolled to cylinder then length be circumference and breadth be the height of the cylinder. Henceforth, by using given formula let us find the radius.

${\small{\underline{\boxed{\sf{Circumference \: of \: circle \: = 2 \pi r}}}}}$

$\sf :\implies Circumference \: of \: circle \: = 2 \pi r \\ \\ \sf :\implies 88 \: = 2 \times 3.14 \times r \\ \\ \sf :\implies 88 \: = 6.28 \times r \\ \\ \sf :\implies \dfrac{88}{6.28} = \: r \\ \\ \sf :\implies 14 \: = r \\ \\ \sf :\implies r \: = 14 \: cm \\ \\ \sf :\implies Radius \: is \: 14 \: cm$

~ Now by using the given formula let us find the volume of the cylinder that is formed.

${\small{\underline{\boxed{\sf{Volume \: of \: cylinder \: = \pi r^{2} h}}}}}$

$\sf :\implies Volume \: of \: cylinder \: = \pi r^{2} h \\ \\ \sf :\implies Volume \: of \: cylinder \: = 3.14 \times 14^{2} \times 40 \\ \\ \sf :\implies Volume \: of \: cylinder \: = 3.14 \times 14 \times 14 \times 40 \\ \\ \sf :\implies Volume \: of \: cylinder \: = 3.14 \times 196 \times 40 \\ \\ \sf :\implies Volume \: of \: cylinder \: = 615.44 \times 40 \\ \\ \sf :\implies Volume \: of \: cylinder \: = 24617.6 \: cm^{3}$