A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter (D) of the hemisphere is equal to the edge of the cube. The surface area of the remaining solid is:

Options

A. `1/4 (pi + 24)D^(2)`
B. `piD^(2)`
C. `(pi-40)D`
D. `(pi + 24) (2D)`

Explanation

To solve the problem, we need to find the surface area of the remaining solid after a hemispherical depression is cut out from one face of a cube. Let's break down the steps:

### Step 1: Understand the dimensions
Let the edge length of the cube be \( a \). Since the diameter of the hemisphere is equal to the edge of the cube, we have:
\[ D = a \]
Thus, the radius \( r \) of the hemisphere is:
\[ r = \frac{D}{2} = \frac{a}{2} \]

### Step 2: Calculate the surface area of the cube
The total surface area \( SA \) of a cube is given by the formula:
\[ SA_{\text{cube}} = 6a^2 \]

### Step 3: Calculate the area of the circular face of the hemisphere
The area of the circular face (the base) of the hemisphere that is cut out is:
\[ A_{\text{circle}} = \pi r^2 = \pi \left(\frac{a}{2}\right)^2 = \pi \frac{a^2}{4} \]

### Step 4: Calculate the curved surface area of the hemisphere
The curved surface area \( SA_{\text{hemisphere}} \) of the hemisphere is given by:
\[ SA_{\text{hemisphere}} = 2\pi r^2 = 2\pi \left(\frac{a}{2}\right)^2 = 2\pi \frac{a^2}{4} = \frac{\pi a^2}{2} \]

### Step 5: Calculate the surface area of the remaining solid
When the hemisphere is cut out from the cube, the total surface area of the remaining solid is given by:
\[ SA_{\text{remaining}} = SA_{\text{cube}} - A_{\text{circle}} + SA_{\text{hemisphere}} \]
Substituting the values we calculated:
\[ SA_{\text{remaining}} = 6a^2 - \pi \frac{a^2}{4} + \frac{\pi a^2}{2} \]

### Step 6: Simplify the expression
Now, we need to simplify the expression:
\[ SA_{\text{remaining}} = 6a^2 - \frac{\pi a^2}{4} + \frac{2\pi a^2}{4} \]
\[ SA_{\text{remaining}} = 6a^2 + \frac{\pi a^2}{4} \]
\[ SA_{\text{remaining}} = 6a^2 + \frac{\pi a^2}{4} = a^2 \left( 6 + \frac{\pi}{4} \right) \]

### Final Answer
Thus, the surface area of the remaining solid is:
\[ SA_{\text{remaining}} = a^2 \left( 6 + \frac{\pi}{4} \right) \]