Question

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.

Answer 1

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.