Question

A solid cylinder has total surface area of 462 cm². Its curved surface area is one-third of its its total surface area. Find the volume of cylinder.​

Answer 1

Answer:

539 cm³

Step-by-step explanation:

Given:-

A solid cylinder has total surface area of 462 cm². Its curved surface area is one-third of its its total surface area.

To Find:-

The volume of cylinder

Solution:-

$\rm \: \bigstar \: Total \: surface \: area = 2\pi r(h + r) \: {cm}^{2} \\ \rm \: \bigstar \: Curved \: surface \: area = 2\pi rh \: {cm}^{2} \\ \\ \rm \: We \: have \\ \rm \: CSA = \frac{1}{3} (TSA) \\ \rm \longrightarrow \: 2\pi rh = \frac{1}{3} \bigg[2\pi r(h + r) \bigg] \\ \rm \longrightarrow \: 3(2\pi rh) = 2\pi rh + 2\pi {r}^{2} \\ \rm \longrightarrow6\pi rh = 2\pi rh + 2\pi {r}^{2} \\ \rm \longrightarrow \: 6\pi rh - 2\pi rh = 2\pi {r}^{2} \\ \rm \longrightarrow \: 4\pi rh = 2\pi {r}^{2} \\ \rm \longrightarrow \: \frac{4\pi rh}{2\pi r } = r \\ \rm \longrightarrow2h = r \\ \rm \longrightarrow h = \frac{r}{2} \\ \rm As \: Total \: surface \: area = 462 \\ \\ \therefore \rm \: 2\pi r(h + r) = 462 \\ \rm \implies \: 2\pi \bigg( \frac{r}{2} + r \bigg) = 462 \\ \rm \implies2\pi r \times \frac{3r}{2} = 462 \\ \rm \implies2 \times \frac{22}{7} \times \frac{3}{2} \times {r}^{2} = 462 \\ \rm \implies {r}^{2} = \cancel{\frac{462 \times 7 \times 2}{2 \times 22 \times 3} } \\ \rm \implies {r}^{2} = 49 \\ \rm \implies {r} = \sqrt{49} = 7 \: cm \\ \\ \tt h = \frac{r}{2} = \frac{7}{2} cm$

Now,

$\rm \: Volume \: of \: cylinder = \pi {r}^{2} h \\ \rm \dashrightarrow \: \frac{ \cancel{22}}{ \cancel{7}} \times \cancel{7} \times 7 \times \frac{7}{ \cancel{2}} \\ \rm \dashrightarrow11 \times 7 \times 7 \\ \rm \dashrightarrow \red{ \boxed{ \pink{ \rm \: 539 \: {cm}^{3} }}}$

Answer 2

Given:-

• T.S.A of Cylinder = 462cm²
• C.S.A of Cylinder = 1/3 of TSA

To Find :-

• Volume of Cylinder = ?

Solution :-

To calculate the volume of Cylinder at first we have to find the height of cylinder and it's radius. then calculate volume of Cylinder. To calculate radius and height of cylinder at first we have to assume the radius and height of cylinder be r and h respectively. Simply by applying formula to calculate radius or height of cylinder then calculate it's volume.

Calculation begins :-

⟹ C.S.A of Cylinder = 1/3 × T.S.A of Cylinder

⟹ C.S.A = 462 × 1/3

⟹ C.S.A = 154 cm²

Now calculate radius of cylinder :-

⟹ T.S.A of Cylinder = C.S.A + 2(base area)

⟹ 2πrh + 2πr² = 462

• Putting the value of C.S.A here:-

⟹ 154 + 2πr² = 462

⟹ 2πr² = 462 - 154

⟹ 2 × 22/7 × r² = 312

⟹ 44r² = 308 × 7

⟹ 44r² = 2156

⟹ r² = 49

⟹ r = √49 ⟹ r = 7cm

Now calculate height of cylinder :-

⟹ C.S.A of Cylinder = 2πrh

⟹ 2 × 22/7 × 7 × h = 154

⟹ 44h = 154

⟹ h = 3.5cm

Now calculate the volume of Cylinder :-

⟹ Volume of cylinder = πr²h

⟹ Volume = 22/7 × 7² × 3.5

⟹ Volume = 22 × 7 × 3.5

⟹ Volume = 154 × 3.5

⟹ Volume = 539cm³

Hence,

• The volume of Cylinder = 539 cm³:-