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Problem of the Week
Problem E and Solution
Bug on the Outside

Problem

A ladybug walks on the surface of the \(2\) by \(3\) by \(12\) rectangular prism shown. The ladybug wishes to travel from \(P\) to \(Q\).

The bottom face of the prism has width 3 and length 12. Its height is 2. P is a vertex of the bottom face. Q is the vertex of the top face that is farthest from P (diagonally opposite corner of the prism).

What is the length of the shortest path from \(P\) to \(Q\) that the ladybug could take?

Solution

We fold out the sides of the prism so that they are laying on the same plane as the top of the prism. The diagram below shows the two-dimensional shape that results. As a result of folding out the sides, vertex P of the prism is a vertex of two different faces in the diagram. We call the second instance \(P^{\prime}\). We let \(X\) be the vertex adjacent to \(P\) along the side of length \(3\), and we let \(Y\) be the vertex adjacent to \(P^{\prime}\) along the side of length \(12\).

A shape composed of rectangles. The rectangle at the centre has horizontal length 12, vertical length 3, and top right vertex Q. Along the left edge of the centre rectangle is a rectangle of vertical length 3 and horizontal length 2, with bottom left vertex P and top left vertex X. Along the bottom edge of the centre rectangle is a rectangle of horizontal length 12 and vertical length 2, with bottom left vertex P prime and bottom right vertex Y. There are dashed lines from Q to P and Q to P prime.

The shortest distance for the ladybug to travel is in a straight line from \(P\) to \(Q\) or from \(P^{\prime}\) to \(Q\).

\(PQ\) is the hypotenuse of right-angled triangle \(PXQ\). Using the Pythagorean Theorem, \[PQ^2=PX^2+XQ^2=3^2+14^2=205\] Thus, \(PQ=\sqrt{205}\approx 14.3\), since \(PQ>0\).

\(P^{\prime}Q\) is the hypotenuse of right-angled triangle \(PYQ\). Using the Pythagorean Theorem, \[(P^{\prime}Q)^2=(P^{\prime}Y)^2+YQ^2=12^2+5^2=169\] Thus, \(P^{\prime}Q=13\), since \(P^{\prime}Q>0\).

Since \(P^{\prime}Q<PQ\), the shortest distance for the ladybug to travel is \(13\) units on the surface of the block in a straight line from \(P^{\prime}\) to \(Q\).