A room has dimensions 2.24 m (height) × 5.30 m × 6.55 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) If the fly walks rather than flies, what is the length of the shortest path it can take? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

Question

A room has dimensions 2.24 m (height) × 5.30 m × 6.55 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. (a) What is the magnitude of its displacement? (b) If the fly walks rather than flies, what is the length of the shortest path it can take? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

Answer 1

d=√(5.30^2 + 6.55^2 + 2.24^2) = 8.718

Unfolding the room,

walk along the short wall to a point 1.0018m up the long wall, then up to the opposite corner

walk along the long wall to a point 1.238m up the short wall, then up to the opposite corner.

In either case, the distance traveled is 12.06m

walking the edges, the distance is 14.09m

Answer 2

To find the magnitude of the fly's displacement, we can find the length of the diagonal of the room. We can use the Pythagorean theorem to do this.

(a) Magnitude of Displacement:
The room can be thought of as a rectangular cuboid. The diagonal of the room represents the direct path from one corner to the opposite diagonal corner. We can find the magnitude of the fly's displacement by calculating this diagonal.

Using the Pythagorean theorem, the length of the diagonal (D) can be found using the formula:
D = √(L^2 + W^2 + H^2)

Given:
Height (H) = 2.24 m
Length (L) = 5.30 m
Width (W) = 6.55 m

Plugging in these values into the formula, we get:
D = √(5.30^2 + 6.55^2 + 2.24^2)

Calculating this expression, we find that:
D ≈ 8.33 m

Therefore, the magnitude of the fly's displacement is approximately 8.33 meters.

(b) Length of the Shortest Path:
If the fly walks rather than flies, it wants to find the shortest path to reach the opposite corner. We can imagine unfolding the walls of the room to form a flat plane.

The shortest path can be found by calculating the distance between the two corners in this flattened plane. Since we have a rectangular shape, the shortest path will be along a straight line.

The length (L) of this shortest path can be calculated using the Pythagorean theorem:
L = √(W^2 + H^2)

Given:
Height (H) = 2.24 m
Width (W) = 6.55 m

Plugging in these values into the formula, we get:
L = √(6.55^2 + 2.24^2)

Calculating this expression, we find that:
L ≈ 6.95 m

Therefore, the length of the shortest path the fly can take if it walks instead of flies is approximately 6.95 meters.