A hemispherical depression is cut out from one face of the cubical wooden block such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

To find the surface area of the remaining solid after a hemispherical depression is cut out from a cubical wooden block, we need to follow a few steps.

Step 1: Understand the Dimensions

Let the edge length of the cube be a. The diameter of the hemisphere is equal to the edge of the cube, so the radius r of the hemisphere is:

• r = a / 2

Step 2: Calculate the Surface Area of the Cube

The total surface area of a cube is given by the formula:

• Surface Area of Cube = 6a²

Step 3: Calculate the Surface Area of the Hemisphere

The surface area of a hemisphere (including the flat circular base) is calculated as:

• Surface Area of Hemisphere = 2πr² + πr² = 3πr²

Substituting the radius:

• Surface Area of Hemisphere = 3π(a/2)² = 3π(a²/4) = (3πa²)/4

Step 4: Adjust for the Cut-out

When the hemisphere is cut out, we lose the area of the circular base of the hemisphere from the cube's surface. The area of the circular base is:

• Area of Base = πr² = π(a/2)² = π(a²/4)

Step 5: Calculate the Remaining Surface Area

The surface area of the remaining solid can be found by taking the surface area of the cube, adding the curved surface area of the hemisphere, and subtracting the area of the circular base:

• Remaining Surface Area = Surface Area of Cube - Area of Base + Surface Area of Hemisphere

Substituting the values:

• Remaining Surface Area = 6a² - (πa²/4) + (3πa²/4)

This simplifies to:

• Remaining Surface Area = 6a² + (2πa²/4) = 6a² + (πa²/2)

Final Result

The surface area of the remaining solid is:

• Remaining Surface Area = 6a² + (πa²/2)