Shore Line
March 3rd, 2010
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A woman is in a rowboat 5 miles from shore. She wishes to reach a point 6 miles down the shore line. Whe?
4. A woman is in a rowboat 5 miles from shore. She wishes to reach a point 6 miles down the shore line. Where should she come ashore if she can row 2 mph and walk 4 mph?
Okay, we'll assume there is no current and that what you want to do is minimize time. If she is trying to reach the shore point 6, she will come ashore at point x. She must therefore row the hypotenuse of a right triangle with one leg 5 (the distance from shore) and the other leg x (the distance along the shore), so that's a hypotenuse of length sqrt(25 + x^2) by the Pythagorean theorem. And then she must walk a distance of 6 - x. In terms of time, given the speeds above, she takes t = sqrt(25 + x^2)/2 + (6 - x)/4 hours to reach the point. You need to minimize t.
You can minimize t by putting this function into a graphing calculator if you're allowed to. If this is for calculus, then you're probably supposed to calculate it yourself, and for that you use differentiation; the minimum t occurs where dt/dx = 0. The derivative of the function above is dt/dx = 2x/(4*sqrt(25 + x^2) - 1/4. Setting it equal to zero gives 2x/(4*sqrt(25 + x^2) - 1/4 = 0 ==> 2x/(4*sqrt(25 + x^2) = 1/4 ==> 8x = 4*sqrt(25 + x^2) ==> 64x^2 = 16(25 + x^2) ==> 64x^2 = 400 + 16x^2 ==> 48x^2 = 400 ==> x^2 = 8.333 ==> x = 2.89 miles downshore from her current position. Note that -2.89 is also a solution, and it's meaningful in this problem since the woman could certainly row upshore of her destination, but logic rules it out without even mathematically checking.
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