KA-0082Shortest paths on and around shapes
5 points, difficulty 3 of 3Level 5–6about 165sA room is 8 m long, 4 m wide and 2 m high. An ant walks across the floor and the walls from one bottom corner to the far top corner. What is the shortest distance in metres?
Text description of the figure
The floor of a room, 8 metres by 4 metres, drawn flat, with one 8-metre-by-2-metre wall unfolded upwards from its far edge. A straight line joins the ant's starting corner on the floor to the opposite corner on the unfolded wall.
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Unfold the room flat.
2The strategy
A route across two surfaces becomes a straight line as soon as you flatten them out. Unfold a wall away from the floor, join the two corners with a straight line, and measure it with Pythagoras.
3The full solution
Unfolding the 8 by 2 wall up from the far edge of the 8 by 4 floor makes a flat rectangle 8 across and 4 + 2 = 6 tall. The straight line is the square root of 8 squared plus 6 squared, which is 10.
Solution
The reliable way
Unfold the surfaces the ant will cross. Laying the 8 by 2 wall flat against the far edge of the 8 by 4 floor gives one flat rectangle 8 long and 4 + 2 = 6 high. The ant's route is now a straight line across it, of length the square root of 64 + 36 = 100, which is 10. Other unfoldings give longer lines — about 10.8 and about 12.2 — so 10 is the shortest. The transferable idea: unfold first, then compare every unfolding, because the first one you try is not always the best.
The elegant way
On a box of dimensions a by b by c the shortest surface path is the square root of a squared plus (b + c) squared for the best pairing, and here that is 8 and 4 + 2 — a 6, 8, 10 triangle.
Why this is on the test: Unfolding turns a three-dimensional path into a straight line, and the trap is measuring through the air, which the ant cannot do.