A gigantic tire, with a radius of 100 miles, is rolling down Broadway at 60 mph. One driver fails to notice the tire’s approach until its descending surface is just touching the roof of her car, 6 feet above the road. If she leaves the car immediately and can shrink to within 2 feet of the road’s surface, how long does she have to crawl out of the tire’s path?
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Surprisingly, she has more than 12 seconds. In this diagram, which is not remotely to scale, r = 100 miles, u = 2 feet, and h = 6 feet. The tire moves at v = 88 feet per second. By Pythagoras, a2 = r2 – (r – u)2 = 2ru – u2 and, similarly, b2 = 2rh – h2. The time we seek is x/v, or (b – a)/v, which works out to 12.089356 seconds.
From Dick Hess’ All-Star Mathlete Puzzles, 2009.
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