Question

Find the volume of a right circular cone, the diameter of whose base is 14cm and the area of curved surface is 550cm squared

Answer 1

Answer- The above question is from the chapter 'Surface Areas and Volumes'.

Concept used: For a right circular cone,

1) Curved surface area (C.S.A.) = πrl

2) Volume = $\frac{1}{3} \pi r^{2} h$

3) Diameter = 2 × Radius

4) $l = \sqrt {h^{2} + r^{2}}$

$\implies l^{2} = h^{2} + r^{2}$

Here, r = radius of cone

l = slant height of cone

h = height of cone

Given question: Find the volume of a right circular cone, the diameter of whose base is 14 cm and the area of curved surface is 550 cm².

Solution: For a right circular cone,

Diameter (d) = 14 cm

Radius (r) = d ÷ 2 = 14 ÷ 2 = 7 cm

Curved surface area (C.S.A.) = πrl

550 = πrl

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

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

l = 25 cm

$l^{2} = h^{2} + r^{2}$

$\implies h^{2} = l^{2} - r^{2}$

$h^{2} = 25^{2} - 7^{2}$

$h^{2} = 625 - 49$

$h^{2} = 576$

$h = \sqrt{576}$

h = 24 cm

Volume = $\frac{1}{3} \pi r^{2} h$

Volume = $\frac{1}{3} . \frac{22}{7} . 7^{2} . 24$

Volume = $22 \times 7 \times 8$

Volume = 1232 cm³

∴ Required volume of right circular cone = 1232 cm³.

Answer 2

$\bf{\underline{\underline{\bigstar\bigstar\: Figure : }}}$

$\:\:$

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$\bf{\underline{\underline{\bigstar\bigstar\: Solution : }}}$

$\:\:$

$\footnotesize{ C.S.A \: of \: cone = { 550cm}^{2}}$

$\footnotesize{\implies \pi rl = { 550cm}^{2}}$

$\footnotesize{\implies \dfrac{22}{7} \times \dfrac{14cm}{2} \times l = { 550cm}^{2}}$

$\footnotesize{\implies \dfrac{22}{7} \times 7cm \times l = { 550cm}^{2}}$

$\footnotesize{\implies 22 \times 1cm \times l = { 550cm}^{2}}$

$\footnotesize{\implies 22cm \times l = { 550cm}^{2}}$

$\footnotesize{\implies l = \dfrac{{ 550cm}^{2}}{22cm} }$

$\footnotesize{\implies l = 25cm }$

$\:\:$

$\rule{200}3$

$\:\:$

$\footnotesize{\implies {l}^{2} = {r}^{2} + {h}^{2} }$

$\footnotesize{\implies {l}^{2} - {r}^{2} = {h}^{2} }$

$\footnotesize{\implies {(25cm)}^{2} - {(7cm)}^{2} = {h}^{2} }$

$\footnotesize{\implies {625cm}^{2} - {49cm}^{2} = {h}^{2} }$

$\footnotesize{\implies {576cm}^{2} = {h}^{2} }$

$\footnotesize{\implies \sqrt{{576cm}^{2}} = h}$

$\footnotesize{\implies 24cm = h}$

$\:\:$

$\rule{200}3$

$\:\:$

$\footnotesize{Volume = \pi {r}^{2} \dfrac{h}{3}}$

$\footnotesize{\implies Volume = \dfrac{22}{7} \times {(7cm)}^{2}\times \dfrac{24cm}{3}}$

$\footnotesize{\implies Volume = \dfrac{22}{7} \times {49cm}^{2} \times 8cm}$

$\footnotesize{\implies Volume = 22 \times {7cm}^{2} \times 8cm}$

$\footnotesize{\implies Volume = 22 \times {56cm}^{3} }$

$\footnotesize{\implies Volume = {1232cm}^{3} }$