A circular track has a radius of $60 \mathrm{~m}$. Ali runs around the circular track at a constant speed of $6 \mathrm{~m} / \mathrm{s}$. A track in the shape of an equilateral triangle has a side length of $x \mathrm{~m}$. Darius runs around this triangular track at a constant speed of $5 \mathrm{~m} / \mathrm{s}$. Ali and Darius each complete one lap in exactly the same amount of time. What is the value of $x$ ?
Since the circular track has radius $60 \mathrm{~m}$, its circumference is $2 \pi \cdot 60 \mathrm{~m}$ which equals $120 \pi \mathrm{m}$.

Since Ali runs around this track at a constant speed of $6 \mathrm{~m} / \mathrm{s}$, then it takes Ali $\frac{120 \pi \mathrm{m}}{6 \mathrm{~m} / \mathrm{s}}=20 \pi \mathrm{s}$ to complete one lap.

Since Ali and Darius each complete one lap in the same period of time, then Darius also takes $20 \pi$ s to complete one lap.

Since Darius runs at a constant speed of $5 \mathrm{~m} / \mathrm{s}$, then the length of his track is $20 \pi \mathrm{s} \cdot 5 \mathrm{~m} / \mathrm{s}$ or $100 \pi \mathrm{m}$.

Since Darius's track is in the shape of an equilateral triangle with side length $x \mathrm{~m}$, then its perimeter is $3 x \mathrm{~m}$ and so $3 x \mathrm{~m}=100 \pi \mathrm{m}$ and so $x=\frac{100 \pi}{3}$.

ANSWER: $x=\frac{100 \pi}{3}$