2440.75 – Around the Reservoir

Amy and Felicia stand at opposite each other across a circular reservoir. At a signal, they start racing around the reservoir in opposite directions. The meet for the first time after Felicia has traveled 100 yards and they meet again 60 yards before Amy has completed her first lap. What is the circumference of the circular path around the reservoir?

Solution

Let $c$ be the circumference of the reservoir; $t_1$ be the time of the first meeting; and $t_2$ be the time of the second meeting. Let $r_A$ and $r_F$ be the speeds of the two women. Then because time equals distance over rate,

$t_1 = \frac{\frac{c}{2}-100}{r_A} = \frac{100}{r_F},$

$t_2 = \frac{c+60}{r_A} = \frac{\frac{c}{2}+60}{r_F}.$

These lead to

$\frac{r_F}{r_A} = \frac{100}{\frac{c}{2}-100} = \frac{\frac{c}{2}+60}{c-60}.$

Cross multiplying gives,

$\left( \frac{c}{2}-100 \right) \left(\frac{c}{2}+60\right) = 100 (c-60).$

Simplification yields the quadratic

$c^2 = 480 c,$

so that $c$ must be 480 yards.