Question

A metallic solid cylinder has one end surmounted by a cone of the same radius and a hemisphere, which is also of same radius at that of cone. The conical portion, the cylindercal 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 the total surface area of the three parts obtained.

Answer 1

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 = `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

= `(πr sqrt (h^2+r^2​) ​+ πr^2) + (2πrh + 2πr^2) + 3πr^2`

= `πr sqrt (h^2+r^2) ​+ 2πrh + 6πr^2`

= `πr sqrt (h^2+r^2)+2h+6r)`

Hence, total surface area of the three parts = `πr sqrt (h^2+r^2)+2h+6r)`.