CAPF AC 2023 Q4: General Mental Ability – Quantitative Aptitude
- (a) 847
- (b) 851
- (c) 693
- (d) 654
Answer: (a) 847
Explanation
Goat tied at the middle of a wall
Given: a straight wall is 28 m long; a goat is tied at the middle of the wall with a rope of 21 m; find the area the goat can reach.
How to think: the wall blocks the rope.
On its own side the goat sweeps a half circle.
At each end of the wall, the rope bends round the corner with whatever length is left.
| Region | Radius | Area |
|---|---|---|
| Front of the wall | 21 m | Half circle: (1/2) × (22/7) × 21² = 693 |
| Round the left end | 21 − 14 = 7 m | Half circle: (1/2) × (22/7) × 7² = 77 |
| Round the right end | 21 − 14 = 7 m | Half circle: 77 |
Total: 693 + 77 + 77 = 847 sq. m.
Why 7 m: each end of the wall is 14 m from the middle, so 21 − 14 = 7 m of rope is left there.
Answer: (a) 847.
Why each option is right or wrong
- (a) CORRECT. 693 + 77 + 77 = 847 sq. m.
- (b) Incorrect. 851 is close but does not come from the three half circles.
- (c) Incorrect. 693 is only the half circle in front of the wall; it ignores the two ends.
- (d) Incorrect. 654 is smaller than even the front half circle.
Key point
Tethered animal problems: split the region where an obstacle bends the rope. Remaining rope at a corner = rope length − distance to the corner. Use π = 22/7 when lengths are multiples of 7.
Remember
Half circle of 21 + two half circles of 7 = 847.
Common mistakes
- Stopping at the front half circle of 693 and ignoring the area behind the wall.
- Using a full circle of radius 21, as if the wall were not there.
- Taking the leftover rope as 21 − 28 instead of 21 − 14.
Exam tip
Educated guess: 693 is the obvious first step and it is among the options as a trap. The true answer must be more than 693, and 847 = 693 + 154 uses the neat leftover of 7 m.
Sources
- NCERT Class X Mathematics — Areas Related to Circles (areas of sectors; tethered animal problems)
- R. S. Aggarwal, Quantitative Aptitude — Area