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?
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?
Correct Answer: 10
Approach Solution - 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
Approach Solution -2
This solid-of-revolution question is easier to handle by sorting surfaces into three families: curved surfaces from the outer profile, a curved surface from the hole, and flat surfaces at the two open ends of the 270 degree sweep.
The outer profile has 5 curved parts, so revolving it produces 5 curved surfaces. The hole traces out its own cylindrical surface when revolved, adding 1 more. Because the sweep stops at 270 degrees instead of closing at 360, two flat sectional faces appear where the material begins and ends. Each of these two faces is split by the hole into an outer flat portion and an inner flat portion around the hole edge, giving 4 flat surfaces in total. Adding these families together: 5 curved from the profile, 1 curved from the hole, and 4 flat from the two open ends.
the answer is 10
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