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.


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Answer 1

Explanation:

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Answer 2

$\bf\underline{\underline{\pink{Question:-}}}$

★ 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.

$\bf\underline{\underline{\blue{Solution:-}}}$

let the length of each edge of a given cube be a units.

Then, volume of the cube

= a³ cubic units =V1(say)

It given that the cylinder lies within the cube and touches all its vertical faces.

So, the radius of the base of the cylinder = $\frac{a}{2}$ units and the height of the cylinder = a units

Volume of the cylinder = πr²h

= { $\frac{22}{7}$× $\frac{a²}{2}$×a} cubic units

= $\frac{11a²}{14}$ cubic units V2(says)

A cone is drawn inside the cylinder that both have the same base and same height.

same base same height.

∴ radius of the base of the cone = $\frac{a}{2}$ units

and height of the cone = a units

∴ volume of the cone = ⅓πr²h

= ( $\frac{1}{3}$× $\frac{22}{7}$× $\frac{a²}{4}$×a )

= $\frac{11a³}{42}$ cubic units = V3(say)

∴ ratio of their volume is given as

V1 : V2 : V3 = a³ : $\frac{11a³}{14}$ : $\frac{11a³}{42}$ = 42 : 33 : 11

$\bf\underline{\underline{\green{Extra\: Fomulae:-}}}$

★For a right circular cone of radius = r units, height = h and the slant height = l units, we have

☙Slant height of the cone(l) = √h²+r² units

☙Volume of the cone = ⅓πr²h cubic units

☙Area of curved surface = (πto) sq units = (πr√h²+r²) sq units

☙Total surface area = (area of the curved surface) + (area of base)

= (πrl+πr²) sq units = πr(l+r) sq units

$\bf\underline{\underline{\orange{Extra\: Information:-}}}$

☙The solid generated by the rotation of a right-angled triangle about one of the sides containing the right angle is called a right circular cone.