Four horses are tethered at the four corners of a square plot of side 63…
2025
Four horses are tethered at the four corners of a square plot of side 63 metres, such that they just cannot reach one another. What is the area (in m2) left ungrazed?
- A.
785.8
- B.
780.6
- C.
675.5
- D.
850.5
Attempted by 2 students.
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Correct answer: D
When four animals are tethered at the four corners of a square field with ropes just long enough that neighbouring animals cannot reach each other, each rope length equals half the side of the square. Since each animal grazes a quarter-circle sector of that radius, the four quarter-circles together form exactly one full circle. The ungrazed area is therefore the square's area minus the area of this one circle.
Side of the square, s = 63 m, so each rope (radius) r = s/2 = 63/2 = 31.5 m.
Area of the square = s2 = 632 = 3969 m2.
The four quarter-circle grazing regions combine into one full circle of radius r: grazed area = π r2 = (22/7) × 31.52 = (22/7) × 992.25 = 3118.5 m2.
Ungrazed area = Area of square − grazed area = 3969 − 3118.5 = 850.5 m2.
Cross-check: using π ≈ 3.14 instead of 22/7 gives grazed area ≈ 3115.67 m2 and ungrazed area ≈ 853.3 m2 — close to the 22/7 result, confirming the method; the small gap is only due to the π approximation used.