Question

The surface areas of a sphere and a cube are equal. Show that the ratio of their volumes is `(sqrt(6) : sqrt(π))`.

Answer 1

Given: Let the sphere have radius r and the cube have edge a. Their surface areas are equal: 4πr² = 6a².

Step-wise calculation:

1. From 4πr² = 6a², get `a^2 = (2/3)πr^2`. 

So `a = rsqrt((2π)/3)`.

2. Volumes: `V_"sphere" = (4/3)πr^3`

`V_"cube" = a^3 = r^3 ((2π)/3)^{3/2}`

3. Ratio `V_"sphere": V_"cube" = [(4/3)π r^3]`: 

`[r^3 ((2π)/3)^{3/2}] = ((4/3)π)/((2π)/3)^{3/2}`

4. Simplify: `((2π)/3)^{3/2} = ((2π)/3) xx sqrt((2π)/3)`. 

So the ratio = `(4/3)π xx [3/(2π)] xx 1/sqrt((2π)/3)`

= `2/sqrt((2π)/3)`

5. Continue simplification:

`2/sqrt((2π)/3) = 2 xx sqrt(3)/sqrt(2π)` 

= `2/sqrt(2) xx sqrt(3)/sqrt(π)` 

= `sqrt(2) xx sqrt(3)/sqrt(π)` 

= `sqrt(6)/sqrt(π)`

The ratio of their volumes is `sqrt(6) : sqrt(π)`.