Question

The curved surface area and the total surface area of a cone are 550 cm2 and 704 cm2 respectively. What is the volume of the cone?​

Answer 1

☯ Let's consider radius of cone be r cm.

Given:

$\begin{gathered}\sf Here \begin{cases} & \sf{Total \: surface \: area = \bf{704\: {cm}^{2} }} \\ & \sf{Curved \: surface \: area = \bf{550 \: {cm}^{2} }} \end{cases}\\ \\\end{gathered}$

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$\begin{gathered}\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}\\ \\\end{gathered}$

• The curved surface area and the total surface area of a cone are 550 cm² and 704 cm².

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$\begin{gathered}\dag\;{\underline{\frak{As\;we\;know\;that,}}}\\ \\\end{gathered}$

☯ TSA & CSA of cone is given by,

$\begin{gathered}\star\;{\boxed{\sf{\pink{TSA_{\;(cone)} = \pi rl}}}}\end{gathered}$

$\begin{gathered}\star\;{\boxed{\sf{\pink{CSA_{\;(cone)} = \pi rl + \pi {r}^{2} }}}}\\ \\\end{gathered}$

⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀

$\begin{gathered}\dag\;{\underline{\frak{Now,\:Putting\:values,}}}\\ \\\end{gathered}$

$: \implies \sf \pi rl + \pi {r}^{2} + \pi rl = 704 - 550$

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Therefore,

• πrl is cancelled from both sides.

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$:\implies \sf \pi {r}^{2} = 704 - 550$

$\begin{gathered}:\implies\sf \dfrac{22}{7} {r}^{2} = 154\\ \\ \\ :\implies\sf {r}^{2} = \frac{154 \times 7}{22} \\\ \\ \\ :\implies\sf {r}^{2} = \frac{7 \times 7}{1}\\ \\ \\ :\implies\sf {r}^{2} = 49\\ \\ \\ :\implies\sf r = \sqrt{49} \\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{r= 7cm}}}}}\;\bigstar\\ \\\end{gathered}$

$\therefore\:{\underline{\sf{Radius \:of\:the\:cone\:is\: {\textbf{\textsf{7\:cm}}}.}}}$

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$\begin{gathered}\dag\;{\underline{\frak{Finding\:slant\:height \: of \: the\: cone,}}}\\ \\\end{gathered}$

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$\sf \implies\star\;{\boxed{\sf{\pink{TSA_{\;(cone)} = \pi rl}}}}\\ \\ \\ : \implies \sf \frac{22}{7} \times 7 \times l = 550 \\ \\ \\ : \implies \sf22l = 550 \\ \\ \\: \implies \sf l = \frac{550}{22} \\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{l = 25cm}}}}}\;$

$\therefore\:{\underline{\sf{Slant \: height \:of\:the\:cone\:is\: {\textbf{\textsf{25\:cm}}}.}}}$

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$\begin{gathered}\dag\;{\underline{\frak{Finding\:height \: of \: the\: cone,}}}\\ \\\end{gathered}$

☯ By using Pythagorean theorm,

$\bigstar \: \boxed{ \sf \pink{ {l}^{2} = {r}^{2} + {h}^{2} }} \\ \\ \\ \implies \sf {25}^{2} = {7}^{2} + {h}^{2} \\ \\ \\ \implies \sf625 = 49 + {h}^{2} \\ \\ \\ \implies \sf {h}^{2} = 625 - 49 \\ \\ \\ \implies \sf {h}^{2} = 576 \\ \\ \\ \implies \sf h = \sqrt{576} \\ \\ \\ : \implies {\underline{\boxed{\frak{\purple{h = 24cm}}}}}\;$

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$\begin{gathered}\dag\;{\underline{\frak{As\;we\;know\;that,}}}\\ \\\end{gathered}$

☯ Volume of cone is given by,

• $\begin{gathered}\star\;{\boxed{\sf{\pink{Volume_{\;(cone)} = \pi \times {r}^{2} \frac{h}{3} }}}}\\ \end{gathered}$

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Therefore,

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$\begin{gathered}:\implies\sf \dfrac{22}{7} \times {7}^{2} \times \dfrac{24}{3} \\ \\ \\ :\implies\sf {22} \times {7} \times \dfrac{24}{3} \\ \\ \\ :\implies\sf 22 \times 7 \times 8 \\ \\ \\ :\implies\sf 22 \times 56\\ \\ \\ :\implies{\underline{\boxed{\frak{\purple{1232 {cm}^{3} }}}}}\;\bigstar\\ \\\end{gathered}$

$\therefore\:{\underline{\sf{Thus,\:the\:volume\:of\:the\:cone\:is\: {\pmb{\frak{1232 \: cm^3}}}.}}}$

Answer 2

Given :- Curved surface area of a cone is 550cm² and total surface area is 704cm².

To Find :- Volume of the given cone .

Used Formulae :- Curved Surface Area of cone ( CSA ) = πrl

:- Total Surface area of Cone ( TSA ) = πr ( l + r ) .

:- Volume of cone = πr²h/3.

:- Slant height of the cone = l² = r²+h².

Solution :-

Let , height of the cone = h cm

radius of the cone = r cm

slant height of the cone = l cm

TSA of given Cone = 704 cm²

πr ( l + r ) = 704

πrl + πr² = 704

550 + πr² = 704

πr² = 704 - 550

πr² = 154

$\frac{22}{7} \times {r}^{2} = 154$

${r}^{2} = \frac{154 \times 7}{22}$

${r}^{2} = 49$

$r = \sqrt{49}$

$r = 7cm$

Now , Putting r = 7 cm In the Formulae of Curved surface area of the cone .

$\pi \: r \: l \: = 550$

$\frac{22 \times 7 \times l}{7} = 550$

$22 \times l = 550$

$l = \frac{550}{22}$

$l = 25cm$

Now as ,

l² = r² + h²

( 25 ) ² = ( 7 ) ² + h²

h² = ( 25 ) ² - ( 7 ) ²

h² = 625 - 49

h² = 576

h = √576

h = 24 cm

Now ,

Volume ( V ) of the cone = πr²h/3

$v \: = \frac{22 \times 7 \times 7 \times 24}{3 \times 7}$

$v = 154 \times 8$

V = 1232 cm³.