A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

The greatest diameter the hemisphere can have is equal to the side of the cube. This is because the hemisphere is surmounted on the cube, meaning it sits on top of the cube. The hemisphere will fit perfectly on the cube if its diameter is equal to the side of the cube. Therefore, the greatest diameter the hemisphere can have is 7 cm.

The surface area of the solid (the cube and the hemisphere combined) can be calculated as follows:

The total surface area of a cube is given by the formula 6a^2, where a is the side length of the cube. So, the surface area of the cube is 6 * (7 cm)^2 = 294 cm^2.

The surface area of a hemisphere is given by the formula 2πr^2, where r is the radius of the hemisphere. The radius is half the diameter, so in this case, r = 7 cm / 2 = 3.5 cm. So, the surface area of the hemisphere is 2 * π * (3.5 cm)^2 = 77 cm^2.

However, since the hemisphere is surmounted on the cube, one face of the cube is not exposed. So, we subtract the area of one face of the cube (which is a^2 = (7 cm)^2 = 49 cm^2) from the total surface area.

Therefore, the total surface area of the solid is 294 cm^2 (cube) + 77 cm^2 (hemisphere) - 49 cm^2 = 322 cm^2.