Question

If a frustum,having radii of its circular ends as x and 2x and its height as 3x ,is melted to form a cube ,then the side of the cube is of what length

Answer 1

length of cube is x root 7

Answer 2

Answer:

The length of the side of the cube is $\sqrt[3]{7\pi} \cdot x$

Step-by-step explanation:

  Given : a frustum,having radii of its circular ends as x and 2x and its height as 3x ,is melted to form a cube

we have to find the length of the side of the cube.

Volume of cube is given by (side)³

And Volume of frustum = $\frac{\pi}{3}h(R^2+r^2+R\cdot r)$

Where R and r is radius of the circular ends of the frustum.

and h is the height of frustum

Given : R = 2x , r = x and h = 3x

Substitute, we get,

Volume of frustum = $\frac{\pi}{3}\cdot 3x((2x)^2+x^2+2x\cdot x)$

Simplify, we get,

Volume of frustum = ${\pi}\cdot x(4x^2+x^2+2x^2)$

Simplify, we get,

Volume of frustum = $7{\pi}\cdot x^3$

Since frustum is melted to form a cube

Thus, Volume of frustum = Volume of cube

Let side of cube be 'a' then,

$7{\pi}\cdot x^3=a^3$

Taking third root both side, we have,

$a=\sqrt{7{\pi}}\cdot x$

The length of the side of the cube is $a=\sqrt[3]{7\pi} \cdot x$