Shape factor is a dimensionless quantity that relates the shape of an object to its surface area. It is defined as the ratio of the surface area of the object to the surface area of a reference object with the same volume. In this case, we are considering a hemispherical body placed on a flat surface.

The shape factor of a hemispherical body with respect to itself can be calculated by dividing the surface area of the hemispherical body by the surface area of a reference object with the same volume.

Let's consider the reference object as a sphere with the same volume as the hemispherical body. The surface area of a sphere is given by the formula:

Surface Area of Sphere = 4πr^2

Where r is the radius of the sphere.

Now, the surface area of the hemispherical body can be calculated by considering the curved surface area of the hemisphere and the flat surface area at the base.

Curved Surface Area of Hemisphere = 2πr^2
Base Surface Area of Hemisphere = πr^2

Total Surface Area of Hemisphere = Curved Surface Area + Base Surface Area = 3πr^2

Now, the shape factor can be calculated as:

Shape Factor = Surface Area of Hemisphere / Surface Area of Sphere
= (3πr^2) / (4πr^2)
= 0.75

Therefore, the shape factor of a hemispherical body placed on a flat surface with respect to itself is 0.75, which corresponds to option C.

To summarize:

- The shape factor relates the shape of an object to its surface area.
- The shape factor of a hemispherical body with respect to itself is calculated by dividing its surface area by the surface area of a reference object with the same volume.
- In this case, the reference object is a sphere with the same volume as the hemispherical body.
- The surface area of the hemispherical body includes the curved surface area and the flat surface area at the base.
- By calculating the ratio of the surface areas, the shape factor is found to be 0.75.