Question

Q. Find The Slant Height ; CSA and TSA of a Cone Whose Volume is 12936cm^3 and The Diameter of the Base is 42cm.

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Answer 1

see the attachmens....

Answer 2

Answer:

$\frak{Given}\begin{cases}\sf{Volume\:of\:the\:cone=\bf{12936\:cm^3.}}\\\sf{Diameter\:of\:the\:base=\bf{42\:cm.}}\end{cases}$

$\underline{\bigstar\bf To \:Find:-}$

• Slant Height
• CSA
• TSA

$\underbrace{\underline{\bf\bigstar\: Required \:Solution:-}}$

◐ First, using the given volume and the diameter of the base, we would find the value of height.

◑ Then we would find the slant height, using the radius and the height .

◐ Then we would find CSA & TSA using their respective formulas.

◆ Finding the Height of the cone :

We have :

• Volume = 12936 cm³
• Diameter = 42 cm

so,

• Radius = 21 cm

We know :

$\odot\;\boxed{\sf Volume_{\: cone}=\dfrac{1}{3}\pi r^2 h }$

So,

$\displaystyle \colon \rarr \sf \: 12936 \: {cm}^{3} = \frac{1}{ \cancel3} \times \frac{22} { \cancel7} \times \cancel{ 21} \: cm \times 21 \: cm \times h \\ \\ \colon \rarr \sf \: \cancel{12936} \: \cancel {cm}^{3} = \cancel{22} \times 21 \: \cancel {cm}^{2} \times h \\ \\ \colon \rarr \sf \: \cancel{ 588 }\: cm = \cancel{21} \times h \\ \\ \colon \dashrightarrow \boxed { \frak{ \orange{h = 28 \: cm}}}$

◆ Finding the Slant height :

We have :

• Radius = 21 cm
• Height = 28 cm

We know :

$\odot\;\boxed{\sf l=r^2+h^2}$

Here,

• l = slant height
• r = radius
• h = height.

So,

$\colon \rarr \sf \: {l}^{2} = {(21 \: cm)}^{2} + {(28 \: cm)}^{2} \\ \\ \colon \rarr \sf \: {l}^{2} = 441 \: cm {}^{2} + 784 \: cm {}^{2} \\ \\ \colon \rarr \sf \: {l}^{2} = 1225 \: {cm}^{2} \\ \\ \colon \rarr \sf \: l = \sqrt{1225 \: {cm}^{2} } \\ \\ \colon \dashrightarrow \underline{\boxed{ \frak{ \purple{l = 35 \: cm}}}}$

◆ Finding the CSA of the cone :

We have :

• r = 21 cm
• l = 35 cm

We know :

$\odot\;\boxed{\sf CSA_{\:cone}=\pi rl}$

So,

$\colon\rarr\sf\: CSA = \dfrac{22}{\cancel7}\times\cancel{21}\: cm\times 35\:cm\\\\\colon\sf\rarr\: CSA = 22\times 3\times 35\: cm^2\\\\\colon\dashrightarrow\underline{\boxed{\frak{\pink{CSA=2310\:cm^2}}}}$

◆ Finding TSA of the Cone :

We have :

• CSA = πrl = 2310 cm²
• r = 21 cm

We know :

$\odot\;\boxed{\sf TSA _{\: cone}=\pi rl+\pi r^2}$

So,

$\colon\rarr\sf \: TSA = 2310 \: cm² +\dfrac{22}{\cancel7}\times \cancel{21}\:cm\times21\:cm\\\\\sf\colon\rarr \:TSA = 2310\:cm^2 + 22\times 3\times21\: cm^2\\\\\sf\colon\rarr\: TSA = 2310\:cm^2 + 1386 \:cm^2\\\\\colon\dashrightarrow\underline{\boxed{\frak{\red{TSA =3696\:cm^2}}}}$

$\underline{\bf \dag\: Final\:Answer:-}$

• Slant Height = 35 cm

• CSA = 2310 cm²

• TSA = 3696 cm²

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