# Problem of the Week

Problem C and Solution

Two Paths

## Problem

Points \(R\), \(S\), \(T\), \(U\), \(V\), and \(W\) lie in a straight line. There are two curved paths from \(R\) to \(W\). The upper path is a semi-circle with diameter \(RW\). The lower path is made up of five semi-circles with diameters \(RS\), \(ST\), \(TU\), \(UV\), and \(VW\).

It is also known that the distance from \(R\) to \(W\) in a straight line is \(1000~\)m, and \(RS=ST=TU=UV=VW\).

Starting at the same time, John and Betty ride their bicycles along these paths from \(R\) to \(W\). Betty follows the upper path and John follows the lower path. If they bike at the same speed, who will arrive at \(W\) first?

## Solution

The circumference of a circle is found by multiplying its diameter by \(\pi\). To find the circumference of a semi-circle, we divide its circumference by \(2\).

The length of the upper path is equal to half the circumference of a circle with diameter \(1000~\)m. Therefore, the length of the upper path is equal to \(\pi \times 1000\div 2=500\pi\) m. (This is approximately \(1570.8\) m.)

Each of the semi-circles along the lower path have the same diameter. The diameter of each of these semi-circles is \(1000\div 5=200\) m. The length of the lower path is equal to half the circumference of five circles, each with diameter \(200\) m. Therefore, the distance along the lower path is equal to \[5\times (\pi \times 200\div 2)=5\times (100\pi)=500\pi\text{ m}\]

Since both John and Betty bike at the same speed and both travel the same distance, they will arrive at point \(W\) at the same time. The answer to the problem may surprise you.

**Extension:**

If you were to extend the problem so that Betty travels the same route but John travels along a lower path made up of \(100\) semi-circles of equal diameter from \(R\) to \(W\), they would still both travel exactly the same distance, \(500\pi\) m. Check it out!