#
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 \(Q\).

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 \(12\).

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
\(Q\).