Question

A solid created by rotating a planar-shape about an axis is called a "Surface of Revolution". The figure below shows a shape and the axis of revolution which is used to create such a solid. Which of the options would be the correct top view of this solid? 

• A. A
• B. B
• C. C
• D. D

Hint:

When interpreting the top view of a surface of revolution, assume that the lines shown correspond to the major features—the peaks (ridges) and valleys (grooves). The spacing of the lines in the top view reflects the differences in the radii of these features on the original profile.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept: 
This question asks for the top-down view of a 3D solid. The solid is generated by revolving a 2D profile curve around a central axis. The top view will show a series of concentric circles, and the key to solving the problem is understanding how the features of the profile curve translate to the positions of these circles. 
Step 2: Key Formula or Approach: 
The top view will show concentric circles corresponding to the significant radial features of the profile curve. These features are the local maxima (peaks) and local minima (valleys) of the curve's distance from the axis of revolution. The radius of each circle in the top view is equal to the horizontal distance of the corresponding peak or valley from the axis. 
Step 3: Detailed Explanation: 
Let's identify the peaks and valleys on the profile curve and their relative distances (radii) from the axis. 
There is a large peak that defines the maximum radius (outermost circle). Let's call its radius R\textsubscript{max}. 
There is another peak to its left with a slightly smaller radius. 
There is another, smaller peak on the far left. 
There are two valleys between these peaks, which will form circles with smaller radii. 
Now, let's analyze the spacing between these features on the profile curve. The spacing between the circles in the top view will be proportional to the difference in the radii of these features. 

The horizontal distance between the highest peak (R\textsubscript{max}) and the peak next to it is relatively small. Therefore, the two outermost circles in the top view should be close together. 
The distance from the second-highest peak to the next valley is larger. Thus, the next gap between circles should be wider. 
The distance from that valley to the leftmost peak is also relatively large. 
The distance between the leftmost peak and the valley next to it is small again. Therefore, the two innermost circles should be close together. 
This gives us a pattern of spacing from outside to inside: a small gap, a large gap, another large gap, and a small gap. 
Step 4: Comparing with Options: 
(A) Shows evenly spaced circles. Incorrect. 
(B) Shows circles that are densely packed at the outer edge and also densely packed near the center, with wider spacing in between. This perfectly matches our analysis of the peak and valley spacing. 
(C) and (D) Show different spacing patterns that do not match our analysis. 
Therefore, option (B) is the correct representation of the top view.