Problem 8 · AMC 8 Stretch
Stretch
Logic & Word Problems
Geometry & Measurement
considering-extreme-caseslogical-reasoningvisual-representation
It is raining straight down, steadily, with no wind. You need to get from point \(C\) to point \(P\). To stay as dry as possible, should you run, walk slowly, or does it not matter?
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Answer: Run — the front catches the same rain at any speed, but the top of your head catches less the faster you go
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Hint 1 of 4
Split your wetness into two parts: rain landing on the TOP of your head, and rain hitting your FRONT as you move forward. Think about each part separately.
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Hint 2 of 4
Top of your head: the longer you are out in the rain, the more drops land on top. So going faster (less time outside) means less rain on top.
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Hint 3 of 4
Your front: as you move forward you 'sweep up' all the drops in your path. Whether you go fast or slow, you still pass through the same stretch of air from \(C\) to \(P\), so your front meets the same number of drops either way.
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Approach: Considering extreme cases — split the wetness into top and front
- Break the problem into two pieces and think about extremes.
- Top of the head: imagine walking infinitely slowly — you stand in the rain forever and your head gets soaked. Imagine running infinitely fast — you are barely outside, so almost nothing lands on top. So faster means less rain on top.
- Front of the body: as you travel from \(C\) to \(P\), you run into all the raindrops in the space you pass through. That space is the same no matter how fast you cross it, so your front collects the same amount of rain at any speed.
- Putting them together: the front is a tie, and the top favors speed. So running gets you to \(P\) driest overall.
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