A hemispherical depression is cut out from one face of a cuboidal wooden block of edge 21cm such that the diameter of the hemisphere is equal to the edge of the cube. Determine the total surface area of the remaining block

(2646 + 110.25π) cm² ≈ 2992.4 cm²

Step-by-step explanation:

The area of a sphere is 4 times the area of a circle with the same radius. Hence the area of a hemisphere will be 2 times the area of that circle. This means carving a hemispherical depression in the face of the cube will add an area that is equal to the area of the circular hole.

Of course the total surface area of a cube is 6 times the area of one square face. The area of a circle is ...

A = πr² = π (d/2) ² = (π/4) d²

The total surface area of the carved cube is ...

S = 6· (21 cm) ² + (π/4) · (21 cm) ² = (441 cm²) (6 + π/4)

S ≈ 2992.36 cm²

The total surface area of the remaining block is about 2992.4 cm².

(2646 + 110.25π) cm² ≈ 2992.4 cm²