Am I on the right track?

• Mar 10th 2013, 01:51 PM
aprilsauce
Am I on the right track? - Solved.
A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the boat be landed in order to arrive at a town 11 miles down the shore from P in the least time?

She must row a distance of : $\sqrt{x^2 + 16}$
She must walk a distance of : $11-x$

$t=\frac{d}{t}$

The time traveled over water plus the time traveled over land can be written:

$\frac{\sqrt{x^2 + 16}}{3} + \frac{11-x}{4}$

If this is correct, how would I find the derivative of the above function..?
Would this be the derivative?

$T^'$ $(x) = \frac{x}{3\sqrt{x^2 + 16}} - \frac{1}{4}$

Finding a common denomenator:

$\frac{4x-3\sqrt{x^2 + 16}}{12\sqrt{x^2 + 16}}$ ?

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I figured out the solution myself. If anyone is curious to how I solved it, here is what I did:

$4x-3\sqrt{x^2 +16} = 0$
$x = \frac{12}{\sqrt{7}}$

From here I plugged in 0, 11, and $\frac{12}{\sqrt{7}}$ back into my original equation. I found that $\frac{12}{\sqrt{7}}$ was the minimum time that occured at the stationary point. Therefore, she would land the boat $\frac{12}{\sqrt{7}}$ miles down the shore from point P.