Question

Shown below is a cross-section which is revolved 270 degrees around the P-Q axis to create a solid. How many surfaces will the resultant solid have?

Hint:

For revolution problems, always visualize the geometry and count the surfaces that will be formed, including flat and curved ones.

Answer 1

The problem asks us to determine the number of surfaces in the solid formed when the given cross-section is revolved 270 degrees around the P-Q axis. 

Analysis Outer curved surfaces: The cross-section contains multiple distinct curves. When these are revolved, each curve generates a separate curved surface in the solid. 

The diagram shows 5 curved parts, so these will create 5 curved outer surfaces in the resultant solid. 

Flat surfaces along the P-Q axis: - The flat base of the cross-section lying along the P-Q axis forms 1 flat surface at the bottom. 

- Since the revolution is only 270 degrees (not a full 360 degrees), there will also be 1 vertical flat surface at the boundary of the open edge. 

Hole in the cross-section: - The hole present in the cross-section forms a cylindrical surface in the resultant solid after the revolution. - Additionally, the top of the hole forms an inner flat surface. 

Counting the Surfaces 

Adding up all the surfaces: 5 + 1 + 1+ 1  + 1 = 10