Two Cyclists and a Linear Graph
Two cyclists, Ana and Boris, ride along the same straight road in the same direction. Ana starts from kilometre marker 0 and rides at a constant 18 km/h. Boris starts 6 km ahead of Ana (at marker 6) and rides at a constant 12 km/h. Their positions in kilometres after t hours are modelled by linear functions: Ana s_A(t) = 18t, Boris s_B(t) = 6 + 12t. A third rider, Vera, claims that because Boris has a head start, Ana can never catch him. The road has a checkpoint at kilometre 30. Times are measured from the moment both start riding.
1. Find algebraically when and where Ana catches Boris, and interpret the slopes and intercepts of the two graphs.
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Set 18t = 6 + 12t, so 6t = 6, giving t = 1 hour. Position: s = 18 × 1 = 18 km. Check with Boris: 6 + 12 × 1 = 18 km. So Ana catches Boris after 1 hour at kilometre 18.
Graph interpretation: the slope is the speed. Ana's slope 18 is steeper than Boris's 12, so her line rises faster. The intercept is the starting position: Ana's is 0, Boris's is 6. Because Ana's line starts lower but is steeper, the lines must cross exactly once - Vera's claim is false.
2. Determine who reaches the 30 km checkpoint first and by how many minutes.
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Ana: 18t = 30, so t = 30/18 = 5/3 h = 1 h 40 min. Boris: 6 + 12t = 30, so 12t = 24, t = 2 h.
Ana arrives first, by 2 h − 1 h 40 min = 20 minutes.
Note the catch-up happens at 18 km, before the checkpoint, so after t = 1 h Ana is ahead and stays ahead. The order at the checkpoint is consistent with the crossing point.
3. After 3 hours, how far apart are the cyclists? Explain why the gap grows after the catch-up point, using the difference of the two functions.
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Difference d(t) = s_A(t) − s_B(t) = 18t − (6 + 12t) = 6t − 6. At t = 3: d = 18 − 6 = 12 km, so Ana is 12 km ahead.
The difference is itself a linear function with slope 6, which is positive. A positive slope means the gap increases by 6 km every hour after the crossing. Before t = 1 the difference is negative (Ana behind); at t = 1 it is zero; after that it grows steadily. This shows the gap does not stay constant - the faster rider keeps pulling away.