Question

A cylinder is within the cube touching all the vertical faces. A cone is inside the cylinder. If their heights are same with the same base, find the ratio of their volumes.​

Answer 1

$\huge\mathfrak{Bonjour!!}$

$\huge\bold\pink{Solution:-}$

☘ Answer:-

☞ Ratio of their volumes= 42:33:11

☘ Step-by-step explanation:-

Let the length of each edge of the cube be a units. Then,

V1 = Volume of the cube = a³ cubic units.

Since a cylinder is within the cube and it touches all the vertical faces of the cube,

Therefore,

r= Radius of the base of the cylinder = a/2,

h= Height of the cylinder = a

Therefore,

V2 = Volume of the cylinder = πr²h

=> V2 = 22/7 × a²/4 × a cubic units

=> V2 = 11/14 a³ cubic units.

Now, it's given that a cone is drawn inside the cylinder such that it has the same base and same height.

Therefore,

V3 = Volume of the cone = 1/3 πr²h

=> V3 = 1/3 × 22/7 × [a/2]² × a cubic units

=> V3 = 11/42 a³ cubic units.

Hence,

☞ The ratio of their volumes = V1 : V2 : V3

= 11/14 a³ : 11/42 a³ = 42:33:11

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Answer 2

${\large\bf{\mid{\overline{\underline{Given:-}}}\mid}}$

• cylinder is inside cube .
• cone is inside that cylinder ..

$\Large\underline\mathfrak{Question}$

• Ratio of their volume ?

$\Large\bold\star\underline{\underline\textbf{Formula\:used}}$

• Volume of cube = (side)³
• volume of cylinder = πr²h
• volume of cone = 1/3 πr²h
• Height of cylinder = side of cube = height of cone
• Radius of cylinder = Side of cube/2 = radius of cone

$\Large\underline{\underline{\sf{Solution}:}}$

$\textbf{Let the edge of the cube be x met.}$

so,

→ height of cylinder = Height of cone = x (m)

→ Radius of cylinder & cone = (x/2) m

$\green{\textbf{now putting all values we get}}$

→ Volume of cube = x³

→ Volume of cylinder = πr²h = π(x/2)²x = 22x³/28

→ Volume of cone =(1/3) πr²h = (1/3)π(x/2) ²x = 22x³/84

→ Cube : cylinder : cone = x³ : 22x³/28 : 22x³/84

→ 1 : 22/28: 22/84

→ 1 : 11/14: 11/42

$\red{\large\boxed{\bold{42 : 33 : 11}}}$

$\large\underline\textbf{Hope it Helps You.}$