10 Fig. 10 shows the speed of a car, in metres per second, during one minute, measured at 10-second intervals.
\begin{figure}[h]
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\caption{Fig. 10}
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The measured speeds are shown below.
| Time \(( t\) seconds \()\) | 0 | 10 | 20 | 30 | 40 | 50 | 60 |
| Speed \(\left( v \mathrm {~m} \mathrm {~s} ^ { - 1 } \right)\) | 28 | 19 | 14 | 11 | 12 | 16 | 22 |
- Use the trapezium rule with 6 strips to find an estimate of the area of the region bounded by the curve, the line \(t = 60\) and the axes. [This area represents the distance travelled by the car.]
- Explain why your calculation in part (i) gives an overestimate for this area. Use appropriate rectangles to calculate an underestimate for this area.
The speed of the car may be modelled by \(v = 28 - t + 0.015 t ^ { 2 }\).
- Show that the difference between the value given by the model when \(t = 10\) and the measured value is less than \(3 \%\) of the measured value.
- According to this model, the distance travelled by the car is
$$\int _ { 0 } ^ { 60 } \left( 28 - t + 0.015 t ^ { 2 } \right) \mathrm { d } t$$
Find this distance.