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.
Step-by-Step Solution
Key Concept: Use the formula for the curved surface area of a hemisphere (\(2\pi r^{2}\)) and subtract the area of the face from which the hemisphere is removed. The edge of the cube equals the diameter of the hemisphere, so the radius is \(r = \frac{l}{2}\).
1. Identify the given data
- Edge of the cube = \(l\)
- Diameter of the hemisphere = \(l\) \(\Rightarrow\) radius \(r = \frac{l}{2}\)
2. Surface area of the original cube
The cube has 6 equal faces, each of area \(l^{2}\).
$$\text{SA}_{\text{cube}} = 6l^{2}$$
3. Effect of cutting the hemispherical depression
- One face (area \(l^{2}\)) is removed.
- The curved surface of the hemisphere becomes exposed.
- Curved surface area of a hemisphere = \(2\pi r^{2}\).
4. Compute the curved surface area of the hemisphere
$$2\pi r^{2}=2\pi\left(\frac{l}{2}\right)^{2}=2\pi\cdot\frac{l^{2}}{4}=\frac{\pi l^{2}}{2}$$
5. Total surface area of the remaining solid
- Keep the 5 untouched faces of the cube: \(5l^{2}\).
- Add the curved surface of the hemisphere: \(\frac{\pi l^{2}}{2}\).
$$\text{SA}_{\text{remaining}} = 5l^{2}+\frac{\pi l^{2}}{2}=l^{2}\left(5+\frac{\pi}{2}\right)$$
6. Final answer
$$\boxed{\text{Surface area}=l^{2}\left(5+\frac{\pi}{2}\right)}$$
Correct Answer: l^2\left(5+\frac{\pi}{2}\right)