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Question

An ant crawls along the edges of a cube of side length 'a'. It starts at one corner and reaches the diagonally opposite corner. If the ant only crawls along the edges and takes the shortest possible path, what is the ratio of the distance it crawls to its displacement?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option B
## CONCEPT FRAMEWORK
	**Primary Concept**: Distance and Displacement
	**Related Topics**: Geometry, Vector addition
	**Prerequisites**: Basic understanding of cubes, distance and displacement concepts.
## SOLUTION PATHWAY
**Given Information**:
	*  Cube with side length 'a'
	*  Ant starts at one corner and moves to the diagonally opposite corner
	*  Ant moves along the edges of the cube
	*  Ant takes the shortest possible path
**Method & Steps**:
	1. **Shortest Distance Traveled**: 
		Reasoning: The shortest path along the edges to reach the opposite corner involves moving along three edges. Each edge has length 'a'.
		Formula/Rule: Total Distance = Sum of lengths of edges
	2. **Calculate Total Distance**:
		Working: Since the ant moves along three edges, the total distance traveled is a+a+a = 3a.
		Key Point: The path could be along x-axis, y-axis and z-axis.
	3. **Calculate Displacement**: 
		Reasoning: The displacement is the straight-line distance between the start and end points. This is the space diagonal of the cube. 
		Formula/Rule: Displacement = $\sqrt{a^2 + a^2 + a^2}$
	4. **Calculate Displacement Value**:
		Working: Displacement = $\sqrt{3a^2} = a\sqrt{3}$
		Key Point: Displacement is the shortest distance between two points.
	5. **Calculate the Ratio**:
		Reasoning: Ratio = (Distance)/(Displacement)
		Working: Ratio = $\frac{3a}{a\sqrt{3}} = \frac{3}{\sqrt{3}} = \frac{3}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}= \frac{3\sqrt{3}}{3} = \sqrt{3}$
		Result: The ratio of distance to displacement is $\frac{3}{\sqrt{3}}$
		Verification: Simplifying $\frac{3}{\sqrt{3}}$ by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ gives $\sqrt{3}$.
 ## OPTION ANALYSIS
[A] :x: Incorrect. This option represents the displacement and not the ratio.
[B] ✓: Correct. This is the simplified ratio of the distance traveled to the displacement.
[C] :x: Incorrect. It's the inverse of the expected ratio. 
[D] :x: Incorrect. The ratio is not 1.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing distance and displacement. Distance is the actual path length, while displacement is the straight-line distance.
		2.[Error 2]: Incorrectly calculating the space diagonal of the cube.
	**Quick Tips**:
		1.[Helpful shortcut]: In a cube, the shortest path along the edges is always 3 times the side length, and the space diagonal is always side length * sqrt(3).
		2.[Memory aid]: Visualize the ant moving along the edges to find the path and the straight line connecting the two corners for displacement.

NAGA SAI · Answer

$\sqrt{3}$

NAGA SAI · Answer

The correct Option is "$3/\sqrt{3}$"

NAGA SAI · Answer

$\sqrt{3}/3$

NAGA SAI · Answer

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