1. **State the problem:** We want to approximate the energy produced by the engine between $t=5$ seconds and $t=15$ seconds. 2. **Recall the formula:** Power $P(t) = 50t^{1.6} + 8t$ watts, and energy is the integral of power over time: $$E = \int_5^{15} P(t) \, dt$$ 3. **Explain the trapezium rule:** To approximate the integral, divide the interval $[5,15]$ into $n$ equal subintervals of width $\Delta t = \frac{15-5}{n} = \frac{10}{n}$. 4. **Trapezium rule formula:** $$E \approx \frac{\Delta t}{2} \left[P(t_0) + 2\sum_{i=1}^{n-1} P(t_i) + P(t_n)\right]$$ where $t_0=5$, $t_n=15$, and $t_i = 5 + i\Delta t$. 5. **Summation setup:** $$\sum_{i=1}^{n-1} P(t_i) = \sum_{i=1}^{n-1} \left(50 t_i^{1.6} + 8 t_i\right)$$ 6. **Interpretation:** This sum adds the power values at the interior points multiplied by 2, plus the power at the endpoints, then multiplied by $\frac{\Delta t}{2}$ to approximate the total energy. 7. **Summary:** Using the trapezium rule, the energy produced between 5 s and 15 s is approximated by $$E \approx \frac{10}{2n} \left[P(5) + 2\sum_{i=1}^{n-1} P\left(5 + i \frac{10}{n}\right) + P(15)\right]$$ This method approximates the area under the power-time curve, representing the energy produced.