Question

If the height of the cylinder is equal to its diameter and the volume is 58212 cubic cm, then find the CSA and TSA of the cylinder.​

Answer 1

Given information,

If the height of the cylinder is equal to its diameter and the volume is 58212 cubic cm, then find the CSA and TSA of the cylinder.

• Height of cylinder = Diameter of cylinder
• Volume of cylinder = 58212 cubic cm
• C.S.A of cylinder = ?
• T.S.A of cylinder = ?

Let,

• Height = Diameter = 2x

So,

• Radius = Diameter/2 = 2x/2 = x

Using formula,

✪ Volume of cylinder = πr²h ✪

Where,

• π = Pi
• r = radius
• h = height

We have,

• π = 22/7
• r = x
• h = 2x
• Volume = 58212 cubic cm

Putting all values,

➻ 58212 = 22/7 × x² × 2x

➻ 58212 = 22/7 × 2x³

➻ 58212 × 7/22 = 2x³

➻ 2646 × 7 = 2x³

➻ 18522 = 2x³

➻ x³ = 18522/2

➻ x³ = 9261

➻ x = cube root (9261)

➻ x = cube root (21 × 21 × 21)

➻ x = 21

• Henceforth, radius of base of cylinder is 21 cm.

Now,

◐ Height of cylinder = 2x

◐ Height of cylinder = 2 × 21

◐ Height of cylinder = 42 cm

• Henceforth, height of cylinder is 42 cm.

Using formula,

✪ C.S.A of cylinder = 2πrh ✪

Where,

• π = Pi
• r = radius
• h = height

We have,

• π = 22/7
• r = 21 cm
• h = 42 cm

Putting all values,

➻ C.S.A of cylinder = 2 × 22/7 × 21 × 42

➻ C.S.A of cylinder = 2 × 22 × 21 × 6

➻ C.S.A of cylinder = 44 × 126

➻ C.S.A of cylinder = 5544 cm²

• Henceforth, C.S.A of cylinder is 5544 cm².

Using formula,

✪ T.S.A of cylinder = 2πr(r + h) ✪

Where,

• π = Pi
• r = radius
• h = height

We have,

• π = 22/7
• r = 21 cm
• h = 42 cm

Putting all values,

➻ T.S.A of cylinder = 2 × 22/7 × 21(21+42)

➻ T.S.A of cylinder = 2 × 22/7 × 21 × 63

➻ T.S.A of cylinder = 2 × 22 × 21 × 9

➻ T.S.A of cylinder = 44 × 189

➻ T.S.A of cylinder = 8316 cm²

• Henceforth, T.S.A of cylinder is 8316 cm².

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Answer 2

$\pink{ \boxed{\boxed{\begin{array}{cc} \bf \: \to \: let \: for \: the \: cylinder : \\ \\ \hookrightarrow \sf \:height \: \: = h \\ \\ \hookrightarrow \sf \:radius \: \: = r \\ \\ \therefore \sf \: diameter \: \: \: d =2r \\ \\ \\ \red{ \underline{\bf \: given \: \: about \: the \: cylinder \: }} \\ \\ \odot \: \sf \: volume \: \: v =58212 \: cm {}^{3} \\ \\ \odot \sf \: height \: = diameter \\ \\ \sf \implies \: h =d \\ \\ \sf \implies \: \boxed{ \sf \: h = 2r} \: \: \: - - - - (1)\end{array}}}}$

$\underline{ \text{ we \: have \: to \: find \: : }}$

$\orange{ \boxed{\boxed{\begin{array}{cc} \hookrightarrow \sf \:curve \: \: \: surface \: \: \: a rea \: \: \: (CSA) \\ \\ \hookrightarrow \sf \:total \: \: surface \: \: \: area \: \: \: (TSA )\end{array}}}}$

Formula will be used :

$\odot \: \: \: \: \green{ \boxed{\sf \: CSA = 2\pi \: rh}} \\ \\ \odot \: \: \: \green{ \boxed{ \sf \: TSA = 2\pi \: r(r + h)}}$

$\dag \: \: \red{ \boxed{ \sf \: volume \: of \: cylinder \: \: \: v = \pi {r}^{2} h}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{ \underline{ \bf \: solution}}$

According to the question :

$\red{ \boxed{\boxed{\begin{array}{cc} \sf \: volume \: \: v = \: \pi {r}^{2} h \\ \\ \sf \implies \:58212 = \frac{22}{7} \times {r}^{2} \times 2r \\ \\ \sf \implies \: {r}^{3} = \frac{58212 \times 7}{ 22 \times 2} \\ \\ \sf \implies \:r {}^{3} = 9261 \\ \\ \sf \implies \:r = \sqrt[3]{9261} \\ \\ \sf \implies \:r = 21 \: cm \\ \\ \sf \: radius \: \: r = 21 \: cm \\ \\ \\ \therefore \sf \: \: height \: \: h = 2 \times 21 = 42 \: cm \\ \\ \\ \orange{ {\boxed{\begin{array}{cc} \sf \: \: r = 21 \: cm \\ \\ \sf \: h = 42 \: cm\end{array}}}} \end{array}}}}$

Now,

$\blue{ \boxed{\boxed{\begin{array}{cc} \sf \: CSA = 2\pi \: rh \\ \\ = 2 \times \frac{22}{7} \times 21 \times42 \\ \\ = 5544 \: \sf \: \: {cm}^{2} \end{array}}}}$

$\blue{ \boxed{\boxed{\begin{array}{cc} \sf \: TSA = 2\pi \: r(r + h) \\ \\ = 2 \times \frac{22}{7} \times 21(21 + 42) \\ \\ = 8361 \: \sf {cm}^{2} \end{array}}}}$

$\therefore \: \: \orange{ \boxed{\boxed{\begin{array}{cc} \sf \: CSA = 5544 \: {cm}^{2} \\ \\ \sf \: TSA = 8361 \: {cm}^{2} \end{array}}}}$