A metallic solid cylinder has one end surmounted by a cone of the same radius and a hemisphere, which is also of same radius as that of the cone. The conical portion, the cylindrical portion and the hemispherical portion are separated as shown below. If the radius of the given solid is r cm and the height of the cone = the height of the cylinder = h cm, find expression for the total surface area of the three parts obtained.

When the three portions are separated, each becomes a complete solid. So, we add the entire surface area of the cone, the cylinder and the hemisphere.

For the cone :

Radius = r cm and height = h cm.

Slant height (l) = $\sqrt{h^2 + r^2}$ cm.

Total surface area of cone = πrl + πr² = $πr\sqrt{h^2 + r^2}$ + πr².

For the cylinder :

Radius = r cm and height = h cm.

Total surface area of cylinder = 2πrh + 2πr².

For the hemisphere :

Radius = r cm.

Total surface area of hemisphere = 2πr² + πr² = 3πr².

∴ Total surface area of the three parts

$= \left(πr\sqrt{h^2 + r^2} + πr^2\right) + \left(2πrh + 2πr^2\right) + 3πr^2 \\[1em] = πr\sqrt{h^2 + r^2} + 2πrh + 6πr^2 \\[1em] = πr\left(\sqrt{h^2 + r^2} + 2h + 6r\right).$

Hence, total surface area of the three parts = πr$\left(\sqrt{h^2 + r^2} + 2h + 6r\right)$ cm².