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.)
To find the magnitude of the fly's displacement, we can use the Pythagorean theorem in three dimensions.
(a) To calculate the magnitude of the displacement, we need to find the length of the diagonal in the room. We can use the Pythagorean theorem twice: once for the vertical displacement and once for the horizontal displacement.
For the vertical displacement, we have the height of the room, which is given as 2.24 m.
For the horizontal displacement, we need to find the diagonal of a rectangle with sides of 5.30 m and 6.55 m. We can use the Pythagorean theorem:
d^2 = 5.30^2 + 6.55^2
Calculating this, we get:
d^2 = 28.09 + 42.80
d^2 = 70.89
Taking the square root of both sides, we get:
d ≈ 8.42 m
So, the magnitude of the fly's displacement is approximately 8.42 m.
(b) If the fly walks instead of flies, we need to find the length of the shortest path it can take along the unfolded walls. We can think of the room as a rectangular box that has been unfolded into a flat plane.
When we unfold the walls, we get a rectangle with dimensions 5.30 m and 6.55 m. The shortest path from one corner to the diagonally opposite corner on a rectangle is a straight line. Therefore, the length of the shortest path is given by the diagonal of the unfolded rectangle.
We can use the Pythagorean theorem to find the length of this diagonal:
d^2 = 5.30^2 + 6.55^2
Calculating this, we get:
d^2 = 28.09 + 42.80
d^2 = 70.89
Taking the square root of both sides, we get:
d ≈ 8.42 m
So, the length of the shortest path the fly can take is approximately 8.42 m, same as the magnitude of its displacement when flying.