Problem 28 · 2013 Math Kangaroo
Stretch
Ratios, Rates & Proportions
distance-speed-timeevaluate-formula
A car starts at point A and drives on a straight road at 50 km/h. Every hour after that, another car leaves point A with a speed 1 km/h faster than the one before. The last car leaves A 50 hours after the first car and drives at 100 km/h. What is the speed of the car that is leading 100 hours after the start of the first car?
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Answer: C — 75 km/h
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Hint 1 of 2
The car leaving at hour k has speed 50 + k and, by 100 hours, has driven for 100 − k hours.
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Hint 2 of 2
Maximise the distance (50 + k)(100 − k) over k from 0 to 50.
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Approach: maximise distance travelled at t = 100
- The car leaving at hour k (speed 50 + k) has travelled (50 + k)(100 − k) by t = 100.
- This product is largest at k = 25, giving 75 × 75 = 5625.
- That leading car's speed is 75 km/h.
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