A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Surface area of the remaining solid = l²(6 + π/4) = l²/4 × (24 + π)
Explanation
This problem involves finding the surface area of a combination solid. When a hemispherical depression is cut from one face of a cube, we need to account for: (1) the total surface area of the cube, (2) subtracting the circular base area of the hemisphere (since this portion is removed from the cube's face), and (3) adding the curved surface area of the hemisphere (which becomes the new inner surface). This follows the same principle as Example 2 in the textbook context, where the formula 'TSA of cube – base area of hemisphere + CSA of hemisphere' is applied.
Solution Steps
Step 1: Given - Edge of cube = l, Diameter of hemisphere = l, so radius r = l/2
Step 2: Total Surface Area (TSA) of cube = 6 × (edge)² = 6l²
Step 3: Area of circular base of hemisphere = πr² = π(l/2)² = πl²/4
Step 4: Curved Surface Area (CSA) of hemisphere = 2πr² = 2π(l/2)² = πl²/2
Step 5: Surface area of remaining solid = TSA of cube – base area + CSA of hemisphere
Step 6: = 6l² – πl²/4 + πl²/2 = 6l² + πl²/4 = l²(6 + π/4) = l²/4 × (24 + π)