Question

The curved surface area of a cylinder is 176sq.cm and its base area is 38.5 sq.cm. Find the volume of the cylinder

Answer 1

Given :

• Curved Surface Area = $\sf{176 \: cm}^{2}$

• Base Area = $\sf{38.5 \: cm}^{2}$

To Find :

• Volume of the Cylinder

Solution :

We know that a Cylinder has two circular plane ends. Two plane ends are parallel and are known as Bases of the Cylinder.

Hence,Given Base Area refers to the Area of Circle

Let's find radius from the Base Area :

$\bigstar \boxed{ \sf \: Area \: of \: Circle = \pi {r}^{2} }$

$\longrightarrow \sf \: 38.5 {cm}^{2} = \dfrac{22}{7} \times {r}^{2}$

$\longrightarrow \sf \: 38.5 {cm}^{2} = \dfrac{22}{7} {r}^{2}$

$\longrightarrow \sf \: {r}^{2} = \dfrac{38.5 \times 7}{22}$

$\longrightarrow \sf \: {r}^{2} = \cancel\dfrac{269.5}{22}$

$\longrightarrow \sf \: {r}^{2} = 12.25$

$\longrightarrow \sf \: {r}= \sqrt{12.25}$

$\longrightarrow \sf \: {r}= \sqrt{3.5 \times 3.5}$

$\longrightarrow \sf \: {r}= 3.5 \: cm$

Let's find Height from the given Curved Surface Area :

$\bigstar \boxed{ \sf \: Curved \: Surface \: Area = 2\pi \: r \: h }$

$\sf \longrightarrow 176 {cm}^{2} = 2 \times \dfrac{22}{7} \times 3.5 \times h$

$\sf \longrightarrow 176 {cm}^{2} = \cancel \dfrac{154}{7} \times h$

$\sf \longrightarrow 176 {cm}^{2} = 22 \times h$

$\sf \longrightarrow 176 {cm}^{2} = 22 h$

$\sf \longrightarrow h = \cancel\dfrac{176}{22}$

$\sf \longrightarrow h = 8 \: cm$

Let's find Volume now with the obtained values of Radius and Height

$\bigstar \boxed{ \sf \: Volume \: of \: Cylinder = \pi {r}^{2} h }$

$\longrightarrow \sf \: \dfrac{22}{7} \times 3.5 \times 3.5 \times 8$

$\longrightarrow \sf \: \dfrac{77 \times 28}{7}$

$\longrightarrow \sf \: \cancel\dfrac{2156}{7}$

$\longrightarrow \sf \: 308 \: {cm}^{3}$

Hence,Volume of the Cylinder = 308 cm^3

Answer 2

Given :

• Curved surface area of cylinder is 176 cm²
• Area of base → 38.5 cm²

To Find :

• Volume of the cylinder.

Solution :

Area of base of the cylinder is given as 38.5 cm²

Formula :

$\large{\boxed{\sf{\blue{Area\:of\:base\:=\:\pi\:r^2}}}}$

Block in the data,

$\red{\implies}$ $\sf{38.5\:=\:\dfrac{22}{7}\:r^2}$

$\red{\implies}$ $\sf{\cancel\dfrac{38.5}{22}\:\times\:7\:=\:r^2}$

$\red{\implies}$ $\sf{1.75\:\times\:7\:=r^2}$

$\red{\implies}$ $\sf{12.25=r^2}$

$\red{\implies}$ $\sf{r\:=\:\sqrt{12.25}}$

$\red{\implies}$ $\sf{r=3.5}$

$\large{\boxed{\sf{\green{Radius\:of\:the\:cylinder\:=\:r\:=\:3.5\:cm}}}}$

Curved Surface Area :

We have the curved surface area of the cylinder given as 176 cm²

That gives us the hint to use the formula for the CSA of cylinder and proceed further.

Formula :

$\large{\boxed{\sf{\red{CSA\:_{cylinder}\:=\:2\:\pi\:r\:h}}}}$

Where,

• r = radius of the cylinder = 3.5 cm
• h = height of the cylinder
• π = $\sf{\dfrac{22}{7}}$

Block in the data,

$\red{\implies}$ $\sf{176\:=\:2\:\times\:\dfrac{22}{7}\:\times\:3.5\:h}$

$\red{\implies}$ $\sf{176=\dfrac{44}{7}\:\times\:3.5\:\times\:h}$

$\red{\implies}$ $\sf{176=\dfrac{154}{7}\:\times\:h}$

$\red{\implies}$ $\sf{176\:=\:22h}$

$\red{\implies}$ $\sf{\cancel\dfrac{176}{22}\:=h}$

$\red{\implies}$ $\sf{8\:=\:h}$

$\large{\boxed{\sf{\purple{Height\:of\:the\:cylinder\:=\:8\:cm}}}}$

Volume of the cylinder :

We have all the required quantities to calculate the volume of the cylinder. Let's use the formula.

Formula :

$\large{\boxed{\sf{\blue{Volume_{cylinder}\:=\:\pi\:r^2\:h}}}}$

Block in the data,

$\red{\implies}$ $\sf{Volume_{cylinder}\:=\:\dfrac{22}{7}\:\times\:(3.5)^2\:\times\:8}$

$\red{\implies}$ $\sf{Volume_{cylinder}\:=\:\dfrac{22}{7}\:\times\:12.25\:\times\:8}$

$\red{\implies}$ $\sf{Volume_{cylinder}\:=\:\dfrac{22}{7}\:\times\:98}$

$\red{\implies}$ $\sf{Volume_{cylinder}\:=\:\cancel{\dfrac{2156}{7}}}$

$\red{\implies}$ $\sf{Volume_{cylinder}\:=\:308}$

$\large{\boxed{\sf{\pink{Volume\:of\:cylinder\:=\:308\:cm^3}}}}$