Question

The radius and height of a cylinder are in the ratio 11:7. If the curved surface area of the cylinder is 121 square units, then find its height and radius.​

Answer 1

Answer :-

$\:\\\large{\boxed{\sf{Firstly,\;let's\;understand\;the\;concept\;used\;:-}}}$

Here we see that we are given the CSA of Cylinder. Also here the ratio of radius and height of cylinder are given. We see that since they are in ratio, their must a constant with which both are multiplied to make them their original value. Also, here we will take the constant as x and apply them in the formula.

Let's do it !!

_________________________________________________

★ Formula Used :-

$\:\\ \large{\boxed{\sf{CSA\;of\;Cylinder\;\:=\;\:\bf{2 \pi rh}}}}$

$\: \\\large{\boxed{\sf{2\:\times\:\dfrac{22}{7}\:\times\:11x\:\times\:7x\;\;=\;\:\bf{121\;\;cm^{2}}}}}$

_________________________________________________

★ Question :-

The radius and height of a cylinder are in the ratio 11:7. If the curved surface area of the cylinder is 121 square units, then find its height and radius.

_________________________________________________

★ Solution :-

Given,

» Ratio of radius and height = 11 : 7

» CSA of Cylinder = 121 cm²

• Let the constant by which both radius and height should be multiplied be x.

Then,

• Radius of Cylinder = 11x

• Height of Cylinder = 7x

_________________________________________________

~ For the Value of x :-

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;CSA\;of\;Cylinder\;\:=\;\:\bf{2 \pi rh}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;2\:\times\:\dfrac{22}{7}\:\times\:11x\:\times\:7x\;\;=\;\:\bf{121\;\;cm^{2}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;2\:\times\:22\:\times\:11x\:\times\:x\;\;=\;\:\bf{121\;\;cm^{2}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;2\:\times\:22\:\times\:11x\:\times\:x\;\;=\;\:\bf{121\;\;cm^{2}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;484\:x^{2}\;\;=\;\:\bf{121\;\;cm^{2}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;x^{2}\;\;=\;\:\bf{\dfrac{121}{484}\:\;=\:\;0.25}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\:\;\;x\;\;=\;\:\bf{\sqrt{0.25}\:\;=\:\;\underline{\underline{0.5}}}}}$

$\;\\\large{\underline{\rm{Thus,\;value\;of\;x\;is\;\;\boxed{\bf{0.5}}}}}$

_________________________________________________

Then, for the value of Radius and Height :-

$\:\\\sf{\rightarrow\;\:Radius\;of\;cylinder\;=\;\bf{11x}\;=\;11(0.5)\;=\;\underline{\underline{\bf{5.5\;\;cm}}}}$

$\:\\\large{\boxed{\boxed{\tt{Hence,\;\;Radius\;\;of\;\:Cylinder\;\:=\;\;\bf{5.5\;\;cm}}}}}$

$\:\\\sf{\rightarrow\;\:Radius\;of\;cylinder\;=\;\bf{7x}\;=\;3.5(0.5)\;=\;\underline{\underline{\bf{3.5\;\;cm}}}}$

$\:\\ \large{\boxed{\boxed{\tt{Hence,\;\;Height\;\;of\;\:Cylinder\;\:=\;\;\bf{3.5\;\;cm}}}}}$

_________________________________________________

★ More to know :-

$\:\\ \sf{\leadsto\;\:TSA\;of\;Cylinder\;\:=\;\:\bf{(2 \pi rh)\:+\:(2 \pi r^{2})}}$

$\:\\ \sf{\leadsto\;\:TSA\;of\;Cone\;\:=\;\:\bf{(\pi rl)\:+\:(\pi r^{2})}}$

$\: \\\sf{\leadsto\;\:CSA\;of\;Cone\;\:=\;\:\bf{2 \pi rl}}$

$\:\\ \sf{\leadsto\;\:Volume\;of\;Cone\;\:=\;\:\bf{\dfrac{1}{3}\:\times\:\pi r^{2} h}}$

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

$\: \\\sf{\leadsto\;\:Volume\;of\;Cube\;\:=\;\:\bf{(Side)^{3}}}$

$\:\\ \sf{\leadsto\;\:Volume\;of\;Cuboid\;\:=\;\:\bf{Length\:\times\:Breadth\:\times\:Height}}$

$\: \\\sf{\leadsto\;\:Volume\;of\;Hemisphere\;\:=\;\:\bf{\dfrac{2}{3}\:\times\:\pi r^{3}}}$

$\: \\\sf{\leadsto\;\:Volume\;of\;Sphere\;\:=\;\:\bf{\dfrac{4}{3}\:\times\:\pi r^{3}}}$

Answer 2

Given:-

• Radius and height of a cylinder are in the ratio 11:7.
• Curved surface area of the cylinder is 121 square units.

⠀

To find:-

• Height and Radius.

⠀

Solution:-

★ Let the,

• Radius of cylinder = 11x
• Height of cylinder = 7x

⠀

★ Using Formula

$\star{\boxed{\sf{\orange{C.S.A~ of~ cylinder = 2\pi rh}}}}$

⠀

$\large{\tt{\longmapsto{2\times \dfrac{22}{\cancel{7}}\times 11x \times \cancel{7}x = 121 cm^2}}}$

⠀

$\large{\tt{\longmapsto{2\times 22\times 11x\times x = 121 cm^2}}}$

⠀

$\large{\tt{\longmapsto{44\times 11x\times x = 121 cm^2}}}$

⠀

$\large{\tt{\longmapsto{484x^2 = 121 cm^2}}}$

⠀

$\large{\tt{\longmapsto{x^2 = \dfrac{121}{484}}}}$

⠀

$\large{\tt{\longmapsto{x^2 = 0.25}}}$

⠀

$\large{\tt{\longmapsto{x = \sqrt0.25}}}$

⠀

$\large{\tt{\longmapsto{x = 0.5}}}$

⠀

Hence, the value of x is 0.5.

⠀

★ Let, find the its height and radius.

⠀

$\large{\tt{\longmapsto{Radius = 11\times 0.5 = 5.5}}}$

⠀

$\large{\tt{\longmapsto{Height = 7\times 0.5 = 3.5}}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬