The figure shows perspective views of the same solid object. Which option shows the count of the total number of surfaces?
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When counting surfaces on a complex 3D object, break it down into simpler geometric components or categorized surfaces (e.g., top, bottom, sides, features). Mentally "paint" each surface to keep track of what you've counted.
Step 1: Understanding the Concept:
This problem requires careful 3D visualization and systematic counting of all the distinct faces or surfaces of a complex object, using multiple views to understand its complete geometry.
Step 2: Detailed Explanation:
To accurately count the surfaces, we can methodically identify each one, ensuring none are missed or double-counted. Let's categorize the surfaces by their orientation and location:
Bottom Surface: There is one flat surface on which the object rests. (1 surface)
Top-Facing Surfaces: There are three distinct surfaces facing upwards: the main top surface, the top of the raised rectangular block, and the flat top of the cylindrical part. (3 surfaces)
Groove Surfaces: The wavy groove cut into the top consists of three surfaces: the curved bottom of the groove and its two vertical side walls. (3 surfaces)
Cylindrical Feature Surfaces: The cylinder has its top face (already counted), its curved side, the flat underside of its overhang, and the vertical face on the main body that the overhang covers. (3 surfaces)
Raised Block Surfaces: The raised block has its top face (already counted), its front-facing vertical surface, and its right-facing vertical surface. (2 surfaces)
Main Body Side Surfaces: Let's count the remaining vertical and side faces of the main L-shaped body.
The front face of the L's vertical leg. (1 surface)
The back face of the L's vertical leg. (1 surface)
The left side face of the object. (1 surface)
The right side face of the object (below the raised block). (1 surface)
The inner vertical face of the L-corner. (1 surface)
Let's sum up the count from this breakdown:
\[
1 (\text{Bottom}) + 3 (\text{Top}) + 3 (\text{Groove}) + 3 (\text{Cylinder part}) + 2 (\text{Block part}) + 5 (\text{Main Body sides}) = 17 \text{ surfaces}
\]
Step 3: Final Answer:
By systematically identifying and counting each distinct surface of the object, the total count is 17.
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Approach Solution -2
Most errors in this kind of counting come from either missing a hidden face or counting a shared edge as two separate faces, so checking each candidate total against those two common mistakes narrows it down.
15: this total is too low, and matches what you would get if the three groove surfaces, the curved bottom and its two side walls, were mistakenly counted as just one surface instead of three.
20: this total is too high, matching what happens if the corner where the raised block meets the main body is counted twice, once as part of the block and once as part of the main body.
19: this total overcounts by two, matching a mistake where the underside of the cylindrical overhang is counted separately from the vertical face it actually merges into.
17: recounting the object face by face, the bottom, the tops, the groove walls, the cylinder's faces, the block's faces and the remaining side faces of the main body, without repeating any shared edge or corner, gives exactly this total.
A careful recount that avoids both undercounting the groove and double-counting shared corners lands on 17.
