Q5:

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.

Given:

A cubical wooden block with edge length $l$. A hemispherical depression is cut out from one face of the cube such that the diameter of the hemisphere is equal to the edge of the cube ($d = l$).

To Find:

The total surface area of the remaining solid.

Step 1: Identify the components of the surface area.

The total surface area of the remaining solid consists of two parts:

1. The total surface area of the cube.

2. The curved surface area of the hemispherical depression (which is added to the total area).

3. The area of the circular top of the hemisphere (which must be subtracted from the cube's face because it is removed/hollowed out).

Step 2: Formulate the mathematical expressions.

Let the edge of the cube be $l$.

Total Surface Area of the cube = $6 \times (\text{edge})^2 = 6l^2$.

The diameter of the hemisphere is $l$, so the radius $r = \frac{l}{2}$.

Curved Surface Area (CSA) of the hemisphere = $2\pi r^2 = 2\pi \left(\frac{l}{2}\right)^2 = 2\pi \left(\frac{l^2}{4}\right) = \frac{\pi l^2}{2}$.

Area of the circular base of the hemisphere (to be subtracted) = $\pi r^2 = \pi \left(\frac{l}{2}\right)^2 = \frac{\pi l^2}{4}$.

Step 3: Calculate the total surface area of the remaining solid.

Total Surface Area = (Total Surface Area of Cube) - (Area of circular base) + (CSA of Hemisphere)

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

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

Substitute $r = \frac{l}{2}$ into the equation:

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

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

Step 4: Simplify the expression.

To add these terms, find a common denominator:

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

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

Final Answer: The total surface area of the remaining solid is $\frac{1}{4}l^2(\pi + 24)$ square units.