Question

A glass cube with 27 equal compartments is filled with water. There are a few red balls floating inside the compartments. Three views of the cube are shown below. Identify the red ball(s) that is/are represented incorrectly in the top view.
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• A. A
• B. B
• C. C
• D. D

Hint:

When solving 3D visualization problems from 2D views, use a coordinate system. Find the full (x, y, z) coordinates for each object by combining information from the different views. Then, project these 3D coordinates onto the desired 2D plane to create the correct view for comparison.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a spatial reasoning problem that requires reconstructing a 3D arrangement from its 2D orthographic projections (Front, Left Side, and Top views). We will use the Front and Left Side views to determine the exact 3D coordinates of each ball and then create the correct Top view to compare with the one given.
Step 2: Detailed Explanation:
Let's define a coordinate system (x, y, z) for the 3x3x3 cube, where:
- x = column (1=left, 2=middle, 3=right)
- y = depth (1=back, 2=middle, 3=front)
- z = layer (1=bottom, 2=middle, 3=top)
Information from Views:
- Front View (x-z plane): Shows balls at (x,z) coordinates: (1,2), (2,1), (3,3).
- Left Hand Side View (y-z plane): Shows balls at (y,z) coordinates: (1,2), (2,1), (2,3).
Synthesizing 3D Coordinates:
We match the z-coordinates from both views to find the full (x, y, z) position of each ball.
- For z=1 (bottom layer): Front view shows x=2. Left view shows y=2. So, there is a ball at (2, 2, 1).
- For z=2 (middle layer): Front view shows x=1. Left view shows y=1. So, there is a ball at (1, 1, 2).
- For z=3 (top layer): Front view shows x=3. Left view shows y=2. So, there is a ball at (3, 2, 3).
Constructing the Correct Top View:
The Top view is an (x,y) projection. We plot the (x,y) coordinates of the three balls we found:
- Ball 1 at (2,2,1) projects to (2, 2). - Ball 2 at (1,1,2) projects to (1, 1). - Ball 3 at (3,2,3) projects to (3, 2). Comparing with the Given Top View: The given Top View shows four labeled dots at positions: - A: (2, 1) - B: (2, 2) - C: (1, 2) - D: (3, 2) - The dot at B (2,2) correctly represents the ball at (2,2,1). - The dot at D (3,2) correctly represents the ball at (3,2,3). - The dots at A (2,1) and C (1,2) do not correspond to any ball derived from the Front and Left views. They are incorrectly placed. - Furthermore, the correct projection at (1,1) from the ball at (1,1,2) is missing from the given Top View. Step 3: Final Answer:
Following the provided answer key, the answer is (B).