Question

14.
The radi of two cylinders are in the ratio 5:7 and their heights are in the ratio 3:5. The ratio
of their curved surface area is

​

Answer 1

$\underline{\bf{\dag} \:\mathfrak{Given\; that\: :}}$⠀⠀

⠀

• The radius of two cylinders are in the ratio of 5:7.

$:\implies\sf r_{1} : r_{2} = 5 : 7$

Also,

• And, their heights are in the ratio of 3:5.

$:\implies\sf h_{1} : h_{2} = 3:5$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

⠀

$\underline{\bf{\dag} \:\mathfrak{As\;we\;know\: that\: :}}$⠀⠀⠀

⠀

$\star\;\boxed{\sf{\pink{CSA_{\;(cylinder)} = 2\pi r h}}}$

⠀

$:\implies\sf CSA_{\:(ratio)} = \bigg(\dfrac{2\pi r_{1} h_{1}}{2\pi r_2 h_2}\bigg) \\\\\\:\implies\sf CSA_{\:(ratio)} = \Bigg(\dfrac{\cancel{\;2}\; \times \cancel{\frac{22}{7}} \times 5 \times 3}{\cancel{\;2}\times \cancel{\frac{22}{7}} \times 7 \times 5} \Bigg) \\\\\\:\implies\sf CSA_{\:(ratio)} = \cancel\dfrac{15}{35}\\\\\\:\implies\sf CSA_{\:(ratio)} = \dfrac{3}{7}\\\\\\:\implies{\underline{\boxed{\sf{\pink{CSA_{\;(ratio)} = 3:7}}}}}\;\bigstar$

⠀

$\therefore{\underline{\sf{Hence, \; the\; ratio \; of \: their \: CSA \; is \; \bf{3:7 }.}}}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

$\qquad\boxed{\underline{\underline{\purple{\bigstar \: \bf\:Formula\:Related\:to\:cylinder\:\bigstar}}}}$

• $\sf Area\:of\:base\:of\:cylinder = \bf{\pi r^2}$

• $\sf Total\:Surface\:area\:of\:cylinder = \bf{2 \pi r(r + h)}$

• $\sf Curved\:Surface\:area\:of\:cylinder = \bf{2 \pi rh}$

• $\sf Volume\:of\:cone = \bf{ \dfrac{1}{3} \times Volume_{\:(cylinder)}}$

Answer 2

${\large{\bold{\rm{\underline{Correct \; Question}}}}}$

★ The radii of two cylinders are in the ratio 5:7 and their heights are in the ratio 3:5. The ratio of their curved surface area is

${\large{\bold{\rm{\underline{Given \; that}}}}}$

★ The radii of two cylinders are in the ratio of 5:7

${\longrightarrow}$ r₁ : r₂ = 5:7

★ Cylinder's heights are in the ratio of 3:5

${\longrightarrow}$ h₁ : h₂ = 3:5

${\large{\bold{\rm{\underline{To \; find}}}}}$

★ The ratio of their curved surface area

${\large{\bold{\rm{\underline{Solution}}}}}$

★ The ratio of their curved surface area = 3:7

${\large{\bold{\rm{\underline{Using \; concept}}}}}$

★ Formula to find curved surface area of the cylinder.

${\large{\bold{\rm{\underline{Using \; formula}}}}}$

★ Curved surface area of the cylinder = 2πrh

${\large{\bold{\rm{\underline{Where,}}}}}$

➝ π is pronounced as pi

➝ The value of π is 22/7 or 3.14

➝ r dentoes radius

➝ h denotes height

${\large{\bold{\rm{\underline{Full \; Solution}}}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \dfrac{2 \pi r_{1} h_{1}}{2 \pi r_{2} h_{2}}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \dfrac{2 \times 3.14 \times 5 \times 3}{2 \times 3.14 \times 7 \times 5}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \dfrac{2 \times 3.14 \times 15}{2 \times 3.14 \times 35}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \dfrac{6.28 \times 15}{6.28 \times 35}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \dfrac{94.20}{219.80}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \: \: \: \: \: \dfrac{3}{7}}}$

${\sf{\mapsto CSA_{(ratio)} \: = \: \: \: \: 3:7}}$

${\frak{Henceforth, \: 3:7 \: is \: ratio \: of \: their \: curved \: surface \: area}}$

${\large{\bold{\rm{\underline{Additional \; knowledge}}}}}$

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

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

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

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

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

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