1. **Problem statement:** We have the speed of a jet at intervals of 5 seconds from 0 to 25 seconds. We want to estimate the length of the runway used by the jet to take off (part a) and then explain if this estimate is an underestimate or overestimate (part b). 2. **Formula and concept:** The length of the runway corresponds to the distance traveled by the jet during acceleration. Distance traveled can be found by calculating the area under the speed-time graph because distance is the integral of speed over time. 3. **Estimating distance using trapezium rule:** Since we have discrete speed values, we approximate the area under the speed-time graph using the trapezium rule: $$\text{Distance} \approx \sum_{i=1}^n \frac{(v_{i-1} + v_i)}{2} \times \Delta t$$ where $v_i$ are speeds at times $t_i$ and $\Delta t = 5$ seconds. 4. **Calculate each trapezium area:** - Between 0s and 5s: $\frac{2 + 5}{2} \times 5 = \frac{7}{2} \times 5 = 17.5$ - Between 5s and 10s: $\frac{5 + 10}{2} \times 5 = \frac{15}{2} \times 5 = 37.5$ - Between 10s and 15s: $\frac{10 + 18}{2} \times 5 = \frac{28}{2} \times 5 = 70$ - Between 15s and 20s: $\frac{18 + 28}{2} \times 5 = \frac{46}{2} \times 5 = 115$ - Between 20s and 25s: $\frac{28 + 42}{2} \times 5 = \frac{70}{2} \times 5 = 175$ 5. **Sum all trapezium areas to get total distance:** $$17.5 + 37.5 + 70 + 115 + 175 = 415 \text{ meters}$$ 6. **Answer to part (a):** The estimated length of the runway used by the jet to take off is **415 meters**. 7. **Part (b) explanation:** Since the jet accelerated smoothly, the speed-time graph is a curve, not straight lines between points. The trapezium rule approximates the area under the curve by straight lines connecting points, which tends to underestimate the area if the curve is concave up (speed increasing at an increasing rate). Here, speed increases more rapidly over time, so the trapezium rule underestimates the true distance. **Therefore, the answer to part (a) is an underestimate of the actual runway length.**