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For my homework, I was given this brainteaser:

You’re sunbathing on the island shown on the map below. The island is six miles from shore at the closest point, and the nearest store is a convenience store seven miles down the beach. If you can row at 4 miles an hour, and you can walk at 5 miles an hour, where should you land to get a bag of tortilla chips in the least possible time? (Ignore tides, currents, and sharks. No fair renting a helicopter.)

I found the equation for the time it takes to get to the store to be:

$\frac{x}{4} + \frac{7-\sqrt{x^2 - 36}}{5}$

I then found the derivative:

$\frac{1}{4} - \frac{2x}{10 \sqrt{x^2-36}}$

I didn't find any zeroes for the derivative inside the domain of the function [6, $\sqrt{85}]$, so I answered that the shortest time would be achieved by rowing straight to the store.

The teacher said the answer was eight. Where did I make a mistake?

edit: Given image: enter image description here

edit: Some closure:

My email to him:

On the brainteaser, the store is only 7 miles away, how can the answer be 8 :P

For my formula, 10 gave the same answer as 8 in the formula the answer uses. i.e., rowing 10 miles lands you 8 miles from the point across from the island. Of course, sqrt(85) is less than 10, so it's only faster if you can then walk 7 - 8 = -1 miles to the store and turn back time a little.

His response:

It assumes you can go back in time.

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    Perhaps you could provide the map or at least describe the geometry of the problem?2012-12-03
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    @espen180, Sure, there you go2012-12-03
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    8 is the answer to a related question: if the store is very far down the beach (or 10 miles down the beach, for instance), how far down the beach should you aim for? If you have the question right, your teacher is simply wrong.2012-12-03
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    There is an error in your derivative calculation. Also if it were me I would not set x to be the distance rowed, but rather the horizontal distance from the landing site to the store. That said your method should work (once you correct the derivative), it is just tougher to interpret2012-12-03
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    @MorganSherman, Actually: http://www.wolframalpha.com/input/?i=derivative+of+x%2F4+%2B+%287-sqrt%28x%5E2-36%29%29%2F52012-12-03
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    @Walkerneo ??? The derivative in your link does not agree with your post.2012-12-03
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    @MorganSherman, I forgot to reduce the 10/2 to 5/1. EDIT: Appologies, I see what you mean now. My work is messy, so the x got lost :P2012-12-03
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    @Wolkerneo You are missing an $x$ in the numerator.2012-12-03
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    @MorganSherman, Yes, sorry, I edited my last comment. I missed the x. Thank you for pointing that out, I edited the question.2012-12-03

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