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?
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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
The cross-section shown has a scalloped top made of a run of rounded bumps, a small circular hole punched through it, and it sits with its base along the P-Q axis. Instead of listing surfaces loosely, it helps to count them by TYPE, one type at a time, and then add.
Curved surfaces from the outer scalloped boundary: each rounded bump and each dip between two bumps changes curve direction, so it forms its own separate curved surface once revolved. Counting the bumps and the dips between them along the boundary gives 7 distinct curved surfaces.
Curved surface from the hole: the circular hole traces out one partial cylindrical surface as it sweeps through 270 degrees, giving 1 more surface.
Flat surfaces from the open revolution: since the shape is revolved only 270 degrees and not the full 360, the solid is left open on two sides. Each open side shows the full 2D cross-section (with its hole) as one flat face, giving 2 flat end faces.
The base of the shape lies exactly along the P-Q axis, so it sweeps out at zero radius and does not add any separate surface.
Adding these up: 7 curved surfaces from the outer boundary, 1 from the hole, and 2 flat end faces gives \[ 7 + 1 + 2 = 10 \] surfaces in total. So the resultant solid has 10 surfaces.
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