Question

Surface area of a sphere and cube are equal find the rate of their volumes

Answer 1

Answer : √6:√π

Solution :
________

Given that : The Surface Area of sphere and cube are equal.

According to the question :

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

Now, ratio of Volume of sphere and cube

$\frac{4}{3} \pi {r}^{3} \div {a}^{3} \\ \\ = > \frac{4}{3} \pi( { \frac{r}{a} })^{3} \\ \\ = > \frac{4}{3} \pi \times ({ \frac{ \sqrt{3} }{ \sqrt{2\pi} } })^{3} \\ \\ = > \frac{4}{3} \pi \times \frac{ \sqrt{3} \times \sqrt{3} \times \sqrt{3} }{ \sqrt{2\pi} \times \sqrt{2\pi} \times \sqrt{2\pi} } \\ \\ = > \frac{4}{3} \pi \times \frac{3}{2\pi} \times \frac{ \sqrt{3} }{ \sqrt{2\pi} } \\ \\ = > \frac{ \sqrt{6} }{ \sqrt{\pi} } \\ \\ = > \sqrt{6 } :\sqrt{\pi}$

So, the ratio between volume of sphere and volume of cube will be √6:√π