# 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!