Calculating Surface Area of Cubical Block with Hemispherical Depression

The problem asks for the surface area of a cubical wooden block after a hemispherical depression is made on one face. The diameter of the hemisphere equals the edge length of the cube.

Understanding the Geometry

• Let the edge length of the cube be denoted by '$l$'.
• The diameter of the hemispherical depression is also '$l$'.
• Therefore, the radius '$r$' of the hemisphere is $\frac{l}{2}$.

Surface Area Components

The total surface area of the remaining solid is calculated as follows:

1. Original Surface Area of the Cube: The cube has 6 faces, each with area $l^2$. Total initial area = $6l^2$.
2. Area Removed: When the hemisphere is cut out from one face, a circular area corresponding to the base of the hemisphere is removed from that face. The area of this circle is $\pi r^2 = \pi \left(\frac{l}{2}\right)^2 = \frac{\pi l^2}{4}$.
3. Area Added: The curved surface area of the hemisphere is exposed and added to the total surface area. The curved surface area of a hemisphere is $2\pi r^2$. So, the added area is $2\pi \left(\frac{l}{2}\right)^2 = 2\pi \frac{l^2}{4} = \frac{\pi l^2}{2}$.

Final Surface Area Calculation

The surface area of the remaining solid is:

Surface Area = (Surface Area of Cube) - (Area of Circular Base Removed) + (Curved Surface Area of Hemisphere)

Surface Area = $6l^2 - \pi r^2 + 2\pi r^2$

Surface Area = $6l^2 + \pi r^2$

Substitute $r = \frac{l}{2}$:

Surface Area = $6l^2 + \pi \left(\frac{l}{2}\right)^2$

Surface Area = $6l^2 + \pi \frac{l^2}{4}$

Factor out $l^2$:

Surface Area = $l^2 \left(6 + \frac{\pi}{4}\right)$

Combine the terms inside the parenthesis:

Surface Area = $l^2 \left(\frac{24 + \pi}{4}\right)$

Surface Area = $\frac{l^2(24 + \pi)}{4}$

Surface Area = $\frac{1}{4}l^2(24 + \pi) \text{cm}^2$.