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प्रश्न
A girl is preparing for her first marathon by running on a straight road. She uses a smartwatch to calculate her running speed at different intervals. The graph (Fig.) depicts her velocity versus time. Estimate the running distance based on the graph.
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उत्तर
The area between the time axis and the velocity-time graph equals the running distance.
Examining the graph’s main features and figuring out each segment’s area:
| Segment | Time (h) | Velocity (Km h−1) |
| Start | 0 ⟶ 0.6 | 7.0 |
| Rising | 0.6 ⟶ 1.6 | 7.0 ⟶ 7.5 |
| Steady | 1.6 ⟶ 3.0 | 7.5 |
| Falling | 3.0 ⟶ 4.6 | 7.5 ⟶ 7.0 |
| Slowing | 4.6 ⟶ 5.6 | 7.0 ⟶ 6.5 |
| End | 5.6 ⟶ 6.6 | 6.5 |
In this case, the area provides the distance in kilometres since the time is in hours and the velocity is in km/h−1.
Running distance is the total of the areas under the velocity-time graph at various time intervals.
= Area of rectangle (0 h ⟶ 0.6 h) + Area of trapezium (0.6 h ⟶ 1.6 h) + Area of rectangle (1.6 h ⟶ 3.0 h) + Area of trapezium (3.0 h ⟶ 4.6 h) + Area of trapezium (4.6 h ⟶ 5.6 h) + Area of rectangle (5.6 h ⟶ 6.6 h)
= `(0.6 - 0) xx 7.0 + 1/2 xx (7.0 + 7.5) xx (1.6 - 0.6) + (3.0 - 1.6) xx 7.5 + 1/2 xx (7.5 + 7.0) xx (4.6 - 3.0) + 1/2 xx (7.0 + 6.5) xx (5.6 - 4.6) + (6.6 - 5.6) xx 6.5`
= 0.6 × 7.0 + 0.5 × 14.5 × 1.0 + 1.4 × 7.5 + 0.5 × 14.5 × 1.6 + 0.5 × 13.5 × 1.0 + 1.0 × 6.5
= 4.2 + 7.25 + 10.5 + 11.6 + 6.75 + 6.5
= 46.8 km
Hence, the estimated running distance is approximately 46.8 km.
