Question

The total surface area of a right circular cone of slant height 20cm is 384πcm².

Calculate:
(i) its radius in cm;
(ii) its volume in cm3, in terms of π.

​

Answer 1

Step-by-step explanation:

Given:-

• The total surface area of a right circular cone  is 384πcm².

• Slant height =  20cm

To Find:-

• The radius

• The Volume in terms of π.

Solution:-

Total surface area of a cone = πr( l + r)

Where, " l " is the slant height and " r " is the radius.

$\sf \implies\pi r(l + r) = 384\pi$

$\sf \implies r(20 + r) = 384$

$\sf \implies {r}^{2} + 20r = 384$

$\sf \implies {r}^{2} + 20r - 384 = 0$

By splitting the middle term:-

$\sf \implies {r}^{2} + 32r - 12r - 384 = 0$

$\sf \implies r(r + 32) - 12(r + 32) = 0$

$\sf \implies (r - 12) (r + 32) = 0$

$\sf \implies r - 12 = 0 \: \: \: (or) \: \: \: \: r + 32= 0$

$\sf \implies r = 12, -32$

Since, radius cannot be negative

$\sf \implies r = 12cm$

$\underline{\boxed{\sf \therefore Radius \: of \: the \: cine = 12cm}}$

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

$\sf \implies h = \sqrt{ {20}^{2} - {12}^{2} }$

$\sf \implies h = \sqrt{400 - 144 }$

$\sf \implies h = \sqrt{256 } = 16$

∴ Height of the cylinder (h) = 16cm

$\sf Volume \: of \: the \: cylinder = \dfrac{1}{3}\pi r^2 h$

$\sf = \dfrac{\pi}{3} \times 12 \times 12 \times 16$

$\sf = \dfrac{2304\pi}{3}$

$\sf = 768\pi$

$\underline{\boxed{\sf \therefore Volume \: of \: the \: cylinder = 768\pi}}$

Answer 2

Given:-

❍Total Surface area of cylinder = 384πcm²

❍Slant height = 20cm

Find:-

❍Radius in cm

❍Volume in cm³ in terms of π

Diagram:-

$\setlength{\unitlength}{1.2mm}\begin{picture}(5,5)\thicklines\put(0,0){\qbezier(1,0)(12,3)(24,0)\qbezier(1,0)(-2,-1)(1,-2)\qbezier(24,0)(27,-1)(24,-2)\qbezier(1,-2)(12,-5)(24,-2)}\put(-0.5,-1){\line(1,2){13}}\put(25.5,-1){\line(-1,2){13}}\multiput(12.5,-1.2)(2,0){7}{\line(1,0){1}}\multiput(12.5,-1.2)(0,4){7}{\line(0,1){2}}\put(16,-0.8){\sf{12cm}}\put(6,10){\sf{16cm}} \put(20,10){\sf{20cm}} \end{picture}$

Solution:-

(i)

we, know that

$\huge{\underline{\boxed{\sf T.S.A \: of \: cone = \pi r(l + r)}}}$

where,

• Slant Height, l = 20cm
• T.S.A of cone = 384πcm²

So,

$\dashrightarrow\sf T.S.A \: of \: cone = \pi r(l + r)$

$\dashrightarrow\sf 384 \pi= \pi \times r(20 + r)$

$\dashrightarrow\sf 384 \pi= \pi r(20 + r)$

$\sf \bigstar divide \: by \: \pi \: both \: sides \bigstar$

$\dashrightarrow\sf \dfrac{384 \pi}{ \pi}= \dfrac{\pi r(20 + r)}{ \pi}$

$\dashrightarrow\sf 384=r(20 + r)$

$\dashrightarrow\sf 384=20r + {r}^{2}$

$\dashrightarrow\sf {r}^{2} + 20r - 384 = 0$

$\sf \bigstar using \: middle - split \: term\bigstar$

$\dashrightarrow\sf {r}^{2} + 20r - 384 = 0$

$\dashrightarrow\sf {r}^{2} + 32r - 12r- 384 = 0$

$\dashrightarrow\sf r(r + 32) - 12(r + 32)= 0$

$\dashrightarrow\sf (r - 12)(r + 32)= 0$

$\begin{gathered}\sf r - 12 = 0 \\ \sf r = 12\end{gathered} \qquad \quad\begin{gathered} \sf r + 32= 0 \\ \sf r = - 32\end{gathered}$

$\: \tiny{ \underline{\sf ignoring \: negative \: value \: -32 \: as \: radius \: can't \: be \: negative}}$

$\small{\therefore{\underline{ \sf radius = 12cm}}}$

____________________________

we, know that

$\huge{\underline{ \boxed{\sf {h}^{2} = {l}^{2} - {r}^{2} }}}$

where,

• Slant Height, l = 20cm
• Radius, r = 12cm

So,

$\dashrightarrow\sf {h}^{2} = {l}^{2} - {r}^{2}$

$\dashrightarrow\sf {h}^{2} = {20}^{2} - {12}^{2}$

$\dashrightarrow\sf {h}^{2} = 400 - 144$

$\dashrightarrow\sf {h}^{2} =256$

$\dashrightarrow\sf h= \sqrt{256}$

$\dashrightarrow\sf h= 16cm$

Now, we know that

$\huge{\underline{ \boxed{\sf Volume \: of \: cone = \dfrac{1}{3} \pi {r}^{2}h}}}$

where,

• Radius, r = 12cm
• Height, h = 16cm

So,

$\mapsto\sf Volume \: of \: cone = \dfrac{1}{3} \pi {r}^{2}h$

$\mapsto\sf Volume \: of \: cone = \dfrac{1}{3} \pi \times {(12)}^{2} \times 16$

$\mapsto\sf Volume \: of \: cone = \dfrac{1}{3} \pi \times 144\times 16$

$\mapsto\sf Volume \: of \: cone = \dfrac{1}{3} \pi \times 2304$

$\mapsto\sf Volume \: of \: cone = \dfrac{2304}{3} \pi$

$\mapsto\sf Volume \: of \: cone = 768\pi {cm}^{3}$

$\small{\therefore{\underline{ \sf volume \: of \: cone = 768 \pi {cm}^{3} }}}$

____________________________

Hence,

• Radius = 12cm
• Volume of Cone = 768π cm³