Question

the surface areas of sphere and a cube are equal. Find the ratio of their volumes.​

Answer 1

Answer:

Let the radius of sphere r and side of cube a .

Given

$4\pi {r}^{2} = 6 {a}^{2} \\ \frac{ {r}^{2} }{ {a}^{ 2} } = \frac{6}{4\pi} \implies \frac{r}{a} = \sqrt{ \frac{3}{2\pi} }$

Now ratio of volume

$\frac{ \frac{4}{3}\pi \: {r}^{3} }{ {a}^{3} } \\\\ = \frac{4\pi}{3} ( \frac{r}{a} ) {}^{3} \\ \\ = \frac{4\pi}{3} \times ( \sqrt{ \frac{3}{2\pi} } ) {}^{3} \\ \\ = \frac{4\pi}{3} \times \frac{3}{2\pi} \sqrt{ \frac{3}{2\pi}} \\ \\ =2 \sqrt{ \frac{3}{2\pi} } = \sqrt{ \frac{6}{\pi} } \approx \frac{138}{100}$

Answer 2

surface area of a cube=6

surface area of a sphere =4π

6=4π

=6/4π

r/a=√3/√2√π

volume of a sphere=4/3π

volume of a cube=

4/3π/

(4π/3) *(r/a)cube

4π/3*√3/√2√π*√3/√2√π*√3/√2√π

=2/√2√π

=√2/√π

therefore the ratio of their volumes is √2:√π