1. **State the problem:** A man travelled along a track in 110 minutes, measured to the nearest 5 minutes. The track length is given as 270 km, with half the track assumed to be measured to the nearest 10 km. 2. **Understanding bounds:** - Time is measured to the nearest 5 minutes, so actual time is between $110 - 2.5 = 107.5$ and $110 + 2.5 = 112.5$ minutes. - Length of the track is 270 km, half measured to nearest 10 km means half length is $135$ km with bounds $135 \pm 5$ km, so between 130 km and 140 km. 3. **Could the actual length of the train be greater than 100 km?** - The other half of the track is measured to the nearest 5 km (from part b), but for part a we use 10 km. - Total track length bounds: first half $130$ to $140$ km, second half $135 \pm 5$ km (assuming same as first half for simplicity), so total length bounds are $130 + 130 = 260$ km minimum and $140 + 140 = 280$ km maximum. 4. **Calculate possible train length:** - The man travelled 110 minutes (about 1.833 hours). - Speed bounds: minimum speed $= \frac{260}{1.875}$ hours (using max time 112.5 min = 1.875 h), maximum speed $= \frac{280}{1.792}$ hours (using min time 107.5 min = 1.792 h). - Calculate minimum speed: $$\frac{260}{1.875} = 138.67\text{ km/h}$$ - Calculate maximum speed: $$\frac{280}{1.792} = 156.46\text{ km/h}$$ 5. **Train length estimation:** - If the train speed is between 138.67 km/h and 156.46 km/h, and the time uncertainty is as above, the train length could be estimated by speed times time uncertainty. - The question asks if the train length could be greater than 100 km, which is unlikely given these speeds and times. 6. **Part (b) Jake's assumption was wrong:** - The track was measured to the nearest 5 km, not 10 km. - This reduces the uncertainty in the track length measurement, making the bounds tighter. - Therefore, the possible actual length of the train would be more accurately estimated, possibly affecting the conclusion in part (a). **Final answer:** - (a) No, the actual length of the train could not have been greater than 100 km based on the given bounds. - (b) Measuring to the nearest 5 km reduces uncertainty, so the train length estimate would be more precise, reinforcing the conclusion in (a).