16. The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a lorry at time \(t\) seconds is modelled by
$$v = 5 \left( \mathrm { e } ^ { 0.1 t } - 1 \right) \sin ( 0.1 t ) , \quad 0 \leq t \leq 30 .$$
- Copy and complete the following table, showing the speed of the lorry at 5 second intervals. Use radian measure for \(0.1 t\) and give your values of \(v\) to 2 decimal places where appropriate.
| \(t\) | 0 | 5 | 10 | 15 | 20 | 25 |
| \(\boldsymbol { v }\) | | 1.56 | 7.23 | 17.36 | | |
- Verify that, according to this model, the lorry is moving more slowly at \(t = 25\) than at \(t = 24.5\).
The distance, \(s\) metres, travelled by the lorry during the first 25 seconds is given by \(s = \int _ { 0 } ^ { 25 } v \mathrm {~d} t\).
- Estimate \(s\) by using the trapezium rule with all the values from your table.