A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

Q.A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

Let the radius be \( r \) units and the height of the cone be \( h \) units.
∴ The slant height of the cone \(=\sqrt{r^2+h^2}\) units.

As per the question, \(2πr^2=πr×\sqrt{r^2+h^2}\)
Or, \(2r=\sqrt{r^2+h^2}\)
Or, \(4r^2=r^2+h^2\)
Or, \(3r^2=h^2\)
Or, \(\cfrac{r^2}{h^2}=\cfrac{1}{3}\)
Or, \(\cfrac{r}{h}=\cfrac{1}{√3}\)
Or, \(r:h=1:√3\)

∴ The ratio of the radius to the height of the cone is \(1:√3\).