A cyclist departs from her home at sunrise and rides for 4 hours. Her distance d(t) (in kilometers) from home at time t hours after departure is recorded every 30 minutes and listed in the table below.
t (h)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
d(t) (km)
0
9
19
28
35
41
46
49
51
Analysts want to argue that at some moment during the ride the cyclist was exactly 30 kilometers from home.
(a) Give two distinct contextual reasons why it is reasonable to model d(t) as continuous for 0≤t≤4.
(b) Using only the values in the table and your justification from part (a), explain why there must exist a time c with 1 h<c<2 h for which d(c)=30 km.
(c) Suppose slight GPS error means each listed distance could be off by up to ±0.5 km. Discuss whether the conclusion in part (b) is still guaranteed under this uncertainty, providing clear reasoning.