Question

The height of a right circular cylinder is 10.5 m. Three times the sum of the areas of its two circular faces is twice the area of the curved surface. Find the volume of the cylinder.​

Answer 1

Given,

• Height of right circular cylinder = 10.5 m

Also,

• Thrice the sum of areas of its two circular faces = Twice the area of its curved surface

We know,

$\small \bull \: \: \text{TSA \: \: of \: \: Cylinder} \rm=2\pi {r}^{}(r + h)$

$\small \bull \: \: \text{LSA \: \: of \: \: Cylinder} \rm= 2\pi rh$

$\rm \bull \: \: \text{Volume \: of \: cylinder} = \pi {r}^{2} h$

• Area of circular face of cylinder = 2πr or πd

For all above mentioned Formulae,

$\mapsto \bf{}\pi = \frac{22}{7} \: \: \: \\ \tiny\mapsto \bf r = radius \: \: of \: \: circular \: \: face \: \: of \: \: right \: \: circular \: \: cylinder \\ \tiny\mapsto \bf h = height \: \: of \: \: right \: \: circular \: \: cylinder \\ \tiny\mapsto \bf d = diameter \: \: of \: \: circular \: \: face \: \: of \: \: right \: \: circular \: \: cylinder$

A.T.Q. --

$\rm{}3 \times \{( {\pi r}^{2}) + ( {\pi r}^{2}) \} = 2 \{2\pi rh \} \\ \to \: \rm6\pi {r}^{2} = 4\pi rh \\ \to \rm6\pi {r}^{2} = 4\pi r(10.5) \\ \to \rm6\pi {r}^{ \cancel2} = 42\pi \cancel{r} \\ \rm \to \: r = \frac{42\cancel\pi}{6\cancel\pi} \: \: = \: \pink7 \: m$

Now,

$\tt{Volume \: \: of \: \: cylinder = \pi {r}^{2}h } \\ \implies \sf \frac{22}{7} \times {(7)}^{2} \times 10.5 \\ \\ \implies \sf \frac{22}{ \cancel7} \times \cancel7 \times 7 \times \frac{105}{10} \\ \\ \implies \sf \frac{1617\cancel0}{1\cancel0} \: \: = \bf \red{1617} \: {m}^{3}$

✝️✝️ Extra Formulae

$\boxed{ \bull \: \: { \text{Diagonal \: of \: cylinder} = \rm \sqrt{ {h}^{2} + {d}^{2} } }}$

$\boxed{ \bull \: \: { \text{Diagonal \: of \: cube} = \rm \sqrt{ 3}\times \text{Edge of cube}}}$

$\boxed{ \bull \: \: { \text{Diagonal \: of \: cuboid} = \rm \sqrt{ {l}^{2} + {b}^{2} + {h}^{2} } }}$

Answer 2

Answer :

➥ The volume of the cylinder = 1617 m³

Given :

➤ Height of the right circular cylinder = 10.5 m

To Find :

➤ Volume of the cylinder = ?

Solution :

First we need to find the radius of the cylinder then after we find volume of the cylinder.

$\tt{: \implies 3(2\pi {r}^{2}) = 2(2\pi rh)}$

$\tt{: \implies 3 \times 2\pi {r}^{2} = 2 \times 2\pi rh}$

$\tt{: \implies 6\pi {r}^{2} = 4\pi rh }$

$\tt{: \implies 3r = 2h}$

$\tt{: \implies r = \dfrac{2}{3}h }$

$\tt{: \implies r = \dfrac{2}{ \cancel{ \: 3 \: }} \times \cancel{10.5}}$

$\tt{: \implies r = 2 \times 3.5}$

$\bf{: \implies \underline{ \: \: \underline{ \green{ \: \: r = 7 \: \: }} \: \: }}$

Now ,

We find volume of the cylinder.

$\tt{: \implies v = \pi {r}^{2} h}$

$\tt{: \implies v = \dfrac{22}{ \cancel{ \: 7 \: }} \times \cancel{ \: 7 \: }\times 7 \times 10.5}$

$\tt{: \implies v = 22 \times 7 \times 10.5}$

$\tt{: \implies v = 154 \times 10.5}$

$\bf{: \implies \underline{ \: \: \underline{ \purple{ \: \: v = 1617 {cm}^{3} \: \: }} \: \: }}$

Hence, the volume of the cylinder is 1617 m³.