Question

19.The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m2.

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Answer 1

Answer:

Diagram :

$\setlength{\unitlength}{1mm}\begin{picture}(5,5)\thicklines\multiput(-0.5,-1)(26,0){2}{\line(0,1){40}}\multiput(12.5,-1)(0,3.2){13}{\line(0,1){1.6}}\multiput(12.5,-1)(0,40){2}{\multiput(0,0)(2,0){7}{\line(1,0){1}}}\multiput(0,0)(0,40){2}{\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\multiput(18,2)(0,32){2}{\sf{42\ cm}}\put(9,17.5){\sf{84\ cm}}\end{picture}$

$\begin{gathered}\end{gathered}$

Given :

• The diameter of a roller is 84 cm and its length is 120 cm.
• It takes 500 complete revolutions to move once over to level a playground.

$\begin{gathered}\end{gathered}$

To Find :

• The area of the playground in m².

$\begin{gathered}\end{gathered}$

Using Formulas :

$\longrightarrow\underline{\boxed{\sf{Radius = \dfrac{Diameter}{2}}}}$

$\longrightarrow\underline{\boxed{\sf{CSA \: of \: cylinder = 2 \pi rh }}}$

$\begin{gathered}\end{gathered}$

Solution :

☼ Finding radius of cylinder by substituting the values in the formula :-

$\longrightarrow{\sf{Radius = \dfrac{Diameter}{2}}}$

$\longrightarrow{\sf{Radius = \dfrac{84}{2}}}$

$\longrightarrow{\sf{Radius = \cancel\dfrac{84}{2}}}$

$\longrightarrow{\sf{Radius =42 \: cm}}$

$\longrightarrow{\underline{\boxed{\sf{\red{Radius =42 \: cm}}}}}$

∴ The radius of cylinder is 42 cm.

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☼ Now, finding the curved surface area of cylinder by substituting the values in the formula :-

$\longrightarrow{\sf{CSA \: of \: cylinder = 2 \pi rh }}$

${\longrightarrow{\sf{CSA \: of \: cylinder = 2 \times \dfrac{22}{7} \times 42 \times 120}}}$

${\longrightarrow{\sf{CSA \: of \: cylinder = 2 \times \dfrac{22}{\cancel{7}} \times \cancel{42} \times 120}}}$

${\longrightarrow{\sf{CSA \: of \: cylinder = 2 \times 22\times 6\times 120}}}$

${\longrightarrow{\sf{CSA \: of \: cylinder = 31680 \: {cm}^{2} }}}$

${\longrightarrow{\underline{\boxed{\sf{\red{CSA \: of \: cylinder = 31680 \: {cm}^{2}}}}}}}$

∴ The CSA of cylinder is 31680 cm².

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☼ Now, finding area of 500 revolutions :-

$\longrightarrow{\sf{Area = 500 \times 31680}}$

$\longrightarrow{\sf{Area = 15840000 \: {cm}^{2} }}$

$\longrightarrow{\sf{Area = 15840000 \times \dfrac{1}{100} \times \dfrac{1}{100}}}$

$\longrightarrow{\sf{Area = 15840000 \times \dfrac{1}{10000}}}$

$\longrightarrow{\sf{Area = \dfrac{15840000}{10000}}}$

$\longrightarrow{\sf{Area = \cancel{\dfrac{15840000}{10000}}}}$

$\longrightarrow{\sf{Area = 1584 \: {m}^{2}}}$

${\longrightarrow{\underline{\boxed{\sf{\red{Area = 1584 \: {m}^{2}}}}}}}$

∴ The area of playground is 1584 m².

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Learn More :

$\boxed{\begin{minipage}{6.2 cm}\bigstar \:\underline{\textbf{Formulae Related to Cylinder :}}\\\\\sf {\textcircled{\footnotesize\textsf{1}}} \:Area\:of\:Base\:and\:top =\pi r^2 \\\\ \sf {\textcircled{\footnotesize\textsf{2}}} \:\:Curved \: Surface \: Area =2 \pi rh\\\\\sf{\textcircled{\footnotesize\textsf{3}}} \:\:Total \: Surface \: Area = 2 \pi r(h + r)\\ \\{\textcircled{\footnotesize\textsf{4}}} \: \:Volume=\pi r^2h\end{minipage}}$

$\rule{300}1.5$

Answer 2

Given :

• Diameter of the roller = 84cm
• Length of the roller = 120cm
• Roller takes 500 complete revolutions to move once over to level playground.

To Find :

• Area of the Playground in m²

Solution :

at first we find ,

the Radius of the Roller :

$: \implies \sf \: Diameter = 2 × Radius$

$: \implies \sf Radius = \frac{Diameter}{2}$

$: \implies \sf Radius = ( \frac{84}{2}) cm$

$: \implies { \boxed {\frak{ \red{ Radius = 42cm}}}} \: \: \bigstar$

Now,

Finding Area of the Roller :

Area of the Roller = Curved Surface Area of Roller

$: \implies \sf \: Area = 2πrh$

$\sf: \implies \: Area = (2 × \frac{22}{7} × 42 × 120) cm²$

$\sf: \implies \: Area = (2 × 22 × 6 × 120) cm²$

$: \implies \sf \: Area = (44 × 720) cm²$

$: \implies{ \boxed {\frak{ \green{ Area \: of \: Roller = 31,680 cm² }}}}$

Finding Area of the Playground :

$\sf: \implies \: Area \: of \: 1 \: revolution = Area \: of \: Roller = 31680cm²$

$\sf: \implies \: Area \: of \: 500 \: revolution = (500 × 31680) cm²$

$: \implies{ \boxed {\frak{ \purple{ Area \: of \: 500 \: revolution = 1,58,40,000 cm²</p><p> }}}}$

Converting the Area into m² :

$\sf: \implies 1,58,40,000 × \frac{1}{100} × \frac{1}{100} m²$

$: \implies \sf \: 1,58,400 × \frac{1}{100} m²$

$: \implies{ \boxed {\frak{ \blue{ 1584 m²}}}}$

Therefore ,

Area of the Playground is 1584m²