A medium question on circles for P5–P6, with a step-by-step solution, diagrams and the answer. A P6 topic. Shaded shapes made of circles and squares are common in the Hua Xia Cup's P6 qualifying round and finals.
The question
A running track has two straights 75 m long and two semicircular bends; the innermost edge has radius 30 m, and each lane is 1 m wide.
Ming is in lane 1 and Keung in lane 6; each runs 1 lap along the inside edge of his own lane.
(Take π as 3.14.)
How many cm further does Keung run than Ming?
Show the hints (2)
💡Hint 1: Which parts of the two runs are equally long? Where can the difference come from?
💡Hint 2: The two bends make a circle. How much bigger is one radius than the other? Circumference = 2 × 3.14 × radius.
Worked solution
Have a go first, then read the solution step by step.
1.The two runners' straights are equally long (75 × 2 m each lap), so they make no difference.
2.The two semicircular bends make one whole circle. Ming's circle has radius 30 m; lane 6 is 5 lanes out from lane 1, so Keung's circle has radius 30 + 5 m.
3.Each lap the gap = 2 × 3.14 × (30 + 5) − 2 × 3.14 × 30 = 2 × 3.14 × 5 m = 2 × 3.14 × 500 cm = 3140 cm.
⚠️Common mistake: thinking you must work out both lanes' lengths first. The difference depends only on how far apart the lanes are (5 m), not on the straights or the bends' radius.
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