Question

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

Answer 1

Edge of wooden block (a) = 21 cm 

Diameter of hemisphere = edge of cube 

Radius = \(\frac{21}2\) = 10.5 cm 

Volume of remaining block = Volume of box – Volume of hemisphere 

= a³ - \(\frac{2}3\)πr³

= (2)³ - \(\frac{2}3\)π(10.5)³

6835.5 cm³ 

Surface area of box = 6a² ………… (1) 

Curved surface area of hemisphere = 2πr³ ……….. (2) 

Area of base of hemisphere = πr²……………….. (3) 

So remaining surface area of box = (1) – (2) + (3) 

= 6a² - πr² 2πr²

= 6(21)² - π(10.5)² + 2π(10.5)²

= 2992.5 cm² 

∴Remaining surface area of box = 2992.5 cm² 

Volume of remaining block = 6835.5 cm³