What is the connection between this question and the one about the 400 m athletics track?
Answer: The curved ends of a running track are semicircles. To find how long each lane is, and so how big the stagger must be, we need the perimeter of a circle. Two semicircular ends together make one full circle of length 2πr.
Step-by-step solution
Idea: “This question” is: what is the perimeter of a circle of radius r? The track question needs exactly this, because lanes differ only on the curved parts.
- A 400 m track has two straight parts and two curved ends. The curved ends are semicircles, and together they make one full circle.1 mark
- The straights are the same for every lane, so the difference between lanes comes only from the circles. Lane of radius r: curved length 2πr. The stagger is the difference 2π(r + w) − 2πr. So answering the track question needs the perimeter of a circle.1 mark
The track’s curved ends are semicircles, so the length of each lane, and the stagger between lanes, are found from the perimeter (circumference) of a circle.
Answer to write in the exam
Curved ends of the track = 2 semicircles = 1 full circle.
Straight parts equal for all lanes.
Difference between lanes = 2π(r + w) − 2πr.
∴ Stagger needs the perimeter of a circle.
Common mistakes that cost marks
- Thinking the straight parts cause the stagger. The straights are equal in all lanes.
- Treating the two ends as two full circles. Each end is a semicircle.
How this can come in the exam
Short answer (2 marks)
A track has two straight parts of 100 m each and two semicircular ends, each of radius 35 m (measured along the inner edge). Find the length of the inner edge. (Use π = 227.)
Show answer
Two semicircles = one circle: 2 × 227 × 35 = 220 m (1 mark). Total = 200 + 220 = 420 m (1 mark).Try one yourself
A path has two straight parts of 50 m and two semicircular ends of radius 14 m. Find its total length. (Use π = 227.)
Show answer
2 × 50 + 2 × 227 × 14 = 100 + 88 = 188 m.
More questions like this
- What happens to the perimeter of a square if we double its side?
- What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter? Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (see the figure)? What do you think?
- What is the value of the C/D ratio? How would you estimate this ratio?
- You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L20D. This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2? It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?
- The Mesopotamian Hexagon-to-Circle comparison (see the figure). Can you see why this shows that π > 3?