Question 16
Using identical cubes, make a solid that gives the following projections:
We need to find the arrangement of identical cubes that produces the given Top, Front, and Side Views for each set. We will define the coordinates of the cubes as (x, y, z), where x is the width (left to right), y is the depth (front to back), and z is the height (bottom to top). We assume the origin (0,0,0) is the bottom-left-front corner of the solid.
Step 1 — Analyzing Set 1 Projections
First, let us list the visible cells for each view:
- Top View (i) shows (x,y) positions: (0,0), (1,0), (2,0), (0,1), (1,1).
- Front View (ii) shows (x,z) positions: (0,0), (1,0), (2,0), (0,1).
- Side View (iii) shows (y,z) positions: (0,0), (1,0), (1,1). The dashed line at (y=0, z=1) means there is no cube at (y=0, z=1).
Now, we determine which cubes (x,y,z) exist. A cube exists at (x,y,z) if its projection appears in all three views. We check all possible (x,y,z) combinations within the maximum dimensions (x from 0 to 2, y from 0 to 1, z from 0 to 1):
- For (x,y) = (0,0):
- (0,0,0): Top (0,0) is present, Front (0,0) is present, Side (0,0) is present. So, (0,0,0) exists.
- (0,0,1): Top (0,0) is present, Front (0,1) is present, Side (0,1) is NOT present. So, (0,0,1) does not exist.
- For (x,y) = (1,0):
- (1,0,0): Top (1,0) is present, Front (1,0) is present, Side (0,0) is present. So, (1,0,0) exists.
- (1,0,1): Top (1,0) is present, Front (1,1) is NOT present. So, (1,0,1) does not exist.
- For (x,y) = (2,0):
- (2,0,0): Top (2,0) is present, Front (2,0) is present, Side (0,0) is present. So, (2,0,0) exists.
- (2,0,1): Top (2,0) is present, Front (2,1) is NOT present. So, (2,0,1) does not exist.
- For (x,y) = (0,1):
- (0,1,0): Top (0,1) is present, Front (0,0) is present, Side (1,0) is present. So, (0,1,0) exists.
- (0,1,1): Top (0,1) is present, Front (0,1) is present, Side (1,1) is present. So, (0,1,1) exists.
- For (x,y) = (1,1):
- (1,1,0): Top (1,1) is present, Front (1,0) is present, Side (1,0) is present. So, (1,1,0) exists.
- (1,1,1): Top (1,1) is present, Front (1,1) is NOT present. So, (1,1,1) does not exist.
- For (x,y) = (2,1): Top (2,1) is NOT present. So, no cubes exist here.
The solid for Set 1 consists of the following 6 cubes:
Step 2 — Analyzing Set 2 Projections
First, let us list the visible cells for each view:
- Top View (iv) shows (x,y) positions: (0,0), (1,0), (0,1).
- Front View (v) shows (x,z) positions: (0,0), (1,0), (0,1).
- Side View (vi) shows (y,z) positions: (0,0), (1,0), (1,1).
Now, we determine which cubes (x,y,z) exist. We check all possible (x,y,z) combinations within the maximum dimensions (x from 0 to 1, y from 0 to 1, z from 0 to 1):
- For (x,y) = (0,0): Top (0,0) is present.
- (0,0,0): Front (0,0) is present, Side (0,0) is present. So, (0,0,0) exists.
- (0,0,1): Front (0,1) is present, Side (0,1) is NOT present. So, (0,0,1) does not exist.
- For (x,y) = (1,0): Top (1,0) is present.
- (1,0,0): Front (1,0) is present, Side (0,0) is present. So, (1,0,0) exists.
- (1,0,1): Front (1,1) is NOT present. So, (1,0,1) does not exist.
- For (x,y) = (0,1): Top (0,1) is present.
- (0,1,0): Front (0,0) is present, Side (1,0) is present. So, (0,1,0) exists.
- (0,1,1): Front (0,1) is present, Side (1,1) is present. So, (0,1,1) exists.
- For (x,y) = (1,1): Top (1,1) is NOT present. So, no cubes exist here.
The solid for Set 2 consists of the following 4 cubes:
Step 3 — Analyzing Set 3 Projections
Let us examine the maximum dimensions shown in each view:
- Top View (vii) shows cubes at y-coordinates 0, 1, and 2. This means the maximum depth of the solid is 2 units.
- Side View (ix) shows cubes at y-coordinates 0 and 1. This means the maximum depth of the solid is 1 unit.
These maximum depths are contradictory. A solid cannot have a maximum depth of 2 units when viewed from the top, but only 1 unit when viewed from the side. This means the given projections for Set 3 are inconsistent.
Therefore, no solid can be made that gives all three projections for Set 3.
Answer
(i) The solid for Set 1 consists of 6 cubes at positions: (0,0,0), (1,0,0), (2,0,0), (0,1,0), (1,1,0), (0,1,1). (ii) The solid for Set 2 consists of 4 cubes at positions: (0,0,0), (1,0,0), (0,1,0), (0,1,1). (iii) No solid can be formed for Set 3 because the Top View (vii) shows a maximum depth of 2 units (y-coordinate 2), while the Side View (ix) shows a maximum depth of 1 unit (y-coordinate 1). These dimensions are inconsistent.
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