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

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:-

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

Now,

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