Question

A Solid Cylinder has total surface area of 462cm^2.Its curved surface area is one-third of its total surface area.Find the volume of the cylinder .​

Answer 1

Answer:

To find: Volume of a cylinder

Given:

Total surface area = 462cm²

2πr(r+h). = 462

=> 2×22/7r(r+h). = 462

=> 44/7 r(r+h) = 462

=> r²+rh = 462 × 7/44

=> r²+rh = 73.5

=> rh = 73.5/r² ------①

Curved surface area. = 1/3 of Total surface area

2πrh = 1/3 × 462

=> 2πrh = 154-------②

=>2 × 22/7 × 73.5/r² = 154

=> 44/7 × 73.5/154 = r²

=> r² = 44/7×73.5/154(cancel)

=> r² = 3

=> (r²)² = (3)²(sq.both sides)

=> r = 9--------③

Putting value of r in equation ② we get,

2πrh = 154

=> 44/7 × 9 × h = 154

=> 9h = 154×7/44

=> h = 154×7/44

9

=> h = 2.7

Volume = πr²h

= 22/7×9²×2.7

= 687.34

Therefore, the volume of the cylinder is 687.34cm³

Answer 2

${\huge{\underline{\underline{\rm{Question:-}}}}}$

Q) A solid Cylinder has Total Surface Area of 462 cm² . It's Curved Surface Area is one third of its Total Surface Area . Find the volume of the cylinder .

${\huge{\underline{\underline{\rm{Answer\checkmark}}}}}$

☆ Concept :

• In these type of Questions , we have to first find the dimensions of radius and height. Then using the Formula , we can easily calculate it's Volume .

• In this question , we would first find the relation between height and Radius .

• After finding the relation , using CSA , we would find the value of either Height or Radius .

• Then will put the value in the first Equation to get both height and radius .

• Then , using the formula to find the volume of cylinder we will get the answer .

☆ Given :

• Total Surface Area (TSA) = 462 cm²
• $\rm \: csa = \frac{1}{3} tsa$

☆ To Find :

• The Volume of the Cylinder .

☆ Solution :

We have ,

$\rm \: csa = \frac{1}{3} tsa$

$\mapsto \rm \: \cancel{2 \pi \: r}h= \frac{1}{3} \times \cancel{2 \pi \: r}(h + r)\:$

$\mapsto \rm \: h = \frac{1}{3} (h + r)$

$\mapsto \rm \: 3h = h + r$

$\mapsto \rm \: r = 3h - h$

$\mapsto \boxed{ \rm \: r = 2h}.....(i)$

Now ,

Using the Formula :

$\rm \: tsa = 2 \pi \: r(h + r)$

from eq (i) ,

$\mapsto \rm \: 462 \: = 2 \pi \times 2h(h + 2h)$

$\mapsto \rm \: \cancel{462} = 4 \times \frac{ \cancel{22}}{7} \times h \times 3h$

$\mapsto \rm \: \cancel{21} = \frac{4}{7} \times \cancel3 {h}^{2}$

$\mapsto \rm \: 7 = \frac{4}{7} {h}^{2}$

$\mapsto \rm \: {h}^{2} = \frac{7 \times 7}{4}$

$\mapsto \boxed{ \rm \: h = \frac{7}{2} \: cm}$

Now , Putting this in eq(i)

$\rm \: r = 2h$

$\mapsto \rm \: r = \cancel2 \times \frac{7}{ \cancel2} \: cm$

$\mapsto \boxed{ \rm \: r = 7 \: cm}$

So ,

We now have

• Radius (r) = 7 cm
• Height (h) = $\dfrac{7}{2}$ cm

Now ,

Using the Formula to find Volume

$\rm \: volume = \pi {r}^{2} h$

$\mapsto \rm \: volume = \frac{ \cancel{22}}{ \cancel7} \times {7}^{2} \times \frac{ \cancel7}{ \cancel2} \: {cm}^{3}$

$\mapsto \rm \: volume = 11 \times 49 \: {cm}^{3}$

$\large \mapsto \boxed{ \rm \: volume = 539 \: {cm}^{3} }$

So , the volume of the Cylinder will be 539 cm³ .

_________________________

☆ Know More :

$\rm1. \: tsa \: of \: cylinder = 2 \pi \: r(h + r)$

$\rm2. \: csa \: of \: cylinder = 2 \pi \: rh$

$\rm3. \: volume \: of \: cylinder = \pi {r}^{2} h$

$\rm4. \: the \: unit \: of \: surface \: area \: is \: in \: the \: form : \\ \rm({unit}^{2} )$

$\rm5. \: the \: unit \: of \: volume \: is \: in \: the \: form : \\ \rm \: ({unit}^{3} )$