Question

A sphere and a cube are of the same height. The ratio of their volumes is
A. 3 : 4
B. 21: 11
C. 4 : 3
D. 11 : 21

Answer 1

A sphere and a cube are of the same height. The ratio of their volumes is 11 : 21

Solution:

Given that,

A sphere and a cube are of the same height

Therefore,

The sphere will have the diameter equal to the side of the cube

Let the length of side of cube be "a"

Then, diameter of sphere = a

We know that,

$Radius = \frac{diameter}{2}\\\\Radius = \frac{a}{2}$

Find volume of cube:

$Volume\ of\ cube = a^3 ----- eqn\ 1$

Find volume of sphere:

$Volume\ of\ sphere = \frac{4}{3} \pi r^3\\\\Volume\ of\ sphere = \frac{4}{3} \pi \times (\frac{a}{2})^3\\\\Volume\ of\ sphere = \frac{4}{3} \pi \times \frac{a^3}{8} ---- eqn\ 2$

Find ratio of volume:

Volume of sphere : volume of cube

$Ratio = \frac{4}{3} \pi \times \frac{a^3}{8} : a^3\\\\Ratio = \frac{ \pi }{6} : 1\\\\Ratio = \frac{22}{7} \times \frac{1}{6} : 1\\\\Ratio = \frac{11}{21} : 1\\\\Ratio = 11 : 21$

Thus ratio of volume of sphere to cube is 11 : 21

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Answer 2

The ratio of their volumes is D. 11 : 21

Step-by-step explanation:

From question, we know that the height of the sphere and cube is equal.

The volume of the sphere is given by the formula:

V₁ = 4/3 π (d/2)³

Let the diameter d of the sphere be x units.

V₁ = 4/3 π (x/2)³

V₁ = 4/3 π (x³/8)

∴ V₁ = πx³/6

The volume of the cube is given by the formula:

V₂ = s³

Let the side s of the cube be x units.

∴ V₂ = x³

Now, the ratio of the volumes be,

V₁/V₂ = (πx³/6)/(x³)

V₁/V₂ = π/6

V₁/V₂ = (22/7)/6

∴ V₁/V₂ = 11/21

Thus, the ratio of the volumes, V₁ : V₂ = 11 : 21