If \(A\) is the total surface area and \(V\) is the volume of a sphere, which of the following is correct?

Q.If \(A\) is the total surface area and \(V\) is the volume of a sphere, which of the following is correct? (a) \(A^3=36πV^2\) (b) \(A^2=4V^2\) (c) \(A^3=36V^2\) (d) \(A^3=36V^3\)

Answer: A

Let the radius of the sphere be \(r\) units.
\(∴ 4πr^2 = A\)
Or, \(r^2 = \cfrac{A}{4π}\)
Or, \((r^2)^3 = (\cfrac{A}{4π})^3\) [Cubing both sides]
Or, \(r^6 = \cfrac{A^3}{64π^3}\)

Again, \(\cfrac{4}{3} πr^3 = V\)
Or, \(r^3 = \cfrac{3V}{4π}\)
Or, \((r^3)^2 = (\cfrac{3V}{4π})^2\)
Or, \(r^6 = \cfrac{9V^2}{16π^2}\)

\(∴ \cfrac{A^3}{64π^3} = \cfrac{9V^2}{16π^2} \)
Or, \(\cfrac{A^3}{4π} = 9V^2\)
Or, \(A^3 = 36πV^2\).