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# A child is drowning.

Best Answer bonanova, 29 March 2013 - 09:33 AM

Spoiler for The time depends on
Go to the full post

9 replies to this topic

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Posted 28 March 2013 - 06:16 PM

At Newport Beach, CA, the long, straight shoreline separates the clear water (blue) from the sand (tan). Mitch, the lifeguard, is at M, 80 meters from the water, and sees a drowning child C, 120 meters from the sand.   The points A and B are 280 meters apart along the shoreline. Mitch can run only 4 m/sec but he can swim 8 m/sec. At what spot on the shoreline AB should Mitch aim in order to reach the child at C as soon as possible?   Compute the minimum time to the nearest tenth of a second; compare it to the times for the paths: MC, MAC, MBC.

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### #2 bonanova

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Posted 28 March 2013 - 11:18 PM

Mitch is quite a swimmer - do you want him running twice as fast, instead?

Either way, there is an answer.

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Vidi vici veni.

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Posted 28 March 2013 - 11:20 PM

Mitch is quite a swimmer - do you want him running twice as fast, instead?
Either way, there is an answer.

No it is hard to run in the sand
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### #4 Yoruichi-san

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Posted 29 March 2013 - 02:31 AM

Spoiler for Minimum...

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Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

### #5 bonanova

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Posted 29 March 2013 - 09:33 AM   Best Answer

Spoiler for The time depends on

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Vidi vici veni.

### #6 dark_magician_92

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Posted 29 March 2013 - 10:03 AM

The calculation part is a headache..........

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### #7 bonanova

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Posted 29 March 2013 - 04:31 PM

The calculation part is a headache..........

Sometimes the fun part challenge is to find the shortest an intuitive path through the messy calculations.
Y-san did the brute force calculation.
I tried all the trig approaches until I rediscovered Snell's law (usually used for optics).

It's a cute puzzle because it ends up with similar triangles on the sand and in the water.
I don't know whether that was because the speed ratio was 2.

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Vidi vici veni.

### #8 bonanova

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Posted 29 March 2013 - 06:37 PM

Spoiler for Minimum...

Spoiler for redoing the math

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Vidi vici veni.

### #9 Yoruichi-san

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Posted 29 March 2013 - 07:59 PM

Online equation solver .  Okay, yes, I'm lazy .

You have a great knowledge of different mathematical principles, but I abhor memorizing things and hence, although the name "Snell's Law" sounds vaguely familiar, I have no memory of what it actually entails.  For me, calculus is very intuitive and finding a max/min via derivative is second nature .

• 0
Women are definitely stronger. We are [Fe]males, after all...

Some of what makes me me is real, some of what makes me me is imaginary...I guess I'm just complex. ;P

<3 BBC's Sherlock, the series and the man. "Smart is the new sexy."

Chromatic Witch links now on my 'About Me' page! Episode 3 is finally here!

When life hands me lemons, I make invisible ink.

### #10 bonanova

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Posted 30 March 2013 - 03:45 AM

Online equation solver .  Okay, yes, I'm lazy .

You have a great knowledge of different mathematical principles, but I abhor memorizing things and hence, although the name "Snell's Law" sounds vaguely familiar, I have no memory of what it actually entails.  For me, calculus is very intuitive and finding a max/min via derivative is second nature .

Our approaches are often complementary. I'm not ready to say that I abhor calculus, but I am old (and tired) enough to call in transferable concepts where they apply if it avoids re-deriving a result. But, this afternoon I did the whole d/dx on this problem, but surrendered when I saw x4 and x3 terms. So I didn't derive x=40 that way, I got it numerically from my Newton-Raphson program, then plugged 40 into the stationary condition just to verify the approaches agree.

I worked in optics long enough for Snell's law to have been like f = ma to me. It just shows the bend in light's path at interfaces that creates the least time path. Just as OP asks here. In optics it's refractive indices, in this puzzle it's running vs swimming, but it's the same effect. But my optics journey was 40 years ago and the connection didn't hit me at first. Full disclosure, Snell's law is an extension of the principle of least action, a fundamental underpinning in several disciplines.