Question

A cubical block of side $14\,$cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid. (Use $\pi=3.14$)

Hint:

When a hemisphere sits perfectly on a cube, the top face is fully covered. Add the hemisphere's CSA to only five faces of the cube.

Answer 1

Step 1: Greatest possible diameter.
The hemisphere is placed on the top face of the cube (a square of side $14$ cm). For it to fit exactly, its circular base must be inscribed in the square. Hence the largest possible diameter equals the side of the square: \[ \boxed{\text{Greatest diameter }=14\text{ cm}}, r=\frac{14}{2}=7\text{ cm}. \]

Step 2: Exposed surface area.
Exposed parts: (i) Curved surface area (CSA) of the hemisphere $=2\pi r^2$, (ii) Five faces of the cube (all except the top, which is covered).
So, \[ S=2\pi r^2+5\cdot(14)^2 =2\cdot 3.14\cdot 7^2+5\cdot 196 =2\cdot 3.14\cdot 49+980 =307.72+980. \] \[ \boxed{S=1287.72\ \text{cm}^2}. \]