# Math Help - Another Stupid Differentiaion Problem

1. ## Another Stupid Differentiaion Problem

Hey guys

Sorry ...here's another differentiation problem i can't solve ><" ...i asked my friends but they have no clue and since it's the holz my teacher is inaccessible

Although an example in the textbook showed a similar question, i can't seem to get the answer with the method shown.

Problem: A man in a boat is 4 km from the nearest point O of a straight beach; his destination is 4 km along the beach from O. If he can row at 4 m/h and walk at 5 km/h, how should he proceed in order toreach his destination in the least possible time?

The textbook answer is: Row direct to destination.

Here's my solution:

$4^2 + x^2 = BC$(Pythagoras' Theorem)
$BC = \sqrt{16 + x^2}$
Time to travel BC $= \frac{\sqrt{16 + x^2}}{4}$
Time to travel CD $= \frac{4-x}{5}$
Total time T $= \frac{\sqrt{16 + x^2}}{4} + \frac{4-x}{5}$
$\frac{dT}{dx} = \frac{x\sqrt{16 + x^2}}{4} - \frac{3}{50}$
For least possible time, $\frac{dT}{dx} = 0$
......
When I work out x, it equals approx. 0.06...which does not make sense at all.

2. Hello, xwrathbringerx!

Your error occured in the only part you didn't show us: solving for $x.$
I'm sure it was a simple Algrbra mistake.

The time function is: . $T \;=\;\frac{\sqrt{x^2+16}}{4} + \frac{4-x}{5}$

Rewrite it: . $T \;=\;\frac{1}{4}(x^2+16)^{\frac{1}{2}} + \frac{4}{5} - \frac{1}{5}x$

Then: . $\frac{dT}{dx} \;=\;\frac{1}{8}(x^2+16)^{\text{-}\frac{1}{2}}\!\cdot\!2x - \frac{1}{5} \;=\;0$

So we have: . $\frac{x}{4\sqrt{x^2+16}} \;=\;\frac{1}{5}$

. . Then: . $5x \;=\;4\sqrt{x^2+16}$

. . Square: . $25x^2 \:=\:16(x^2+16) \quad\Rightarrow\quad 25x^2 + 16x^2 + 256 \quad\Rightarrow\quad 9x^2 \:=\:256$

Therefore:. $x^2 \:=\:\frac{256}{9} \quad\Rightarrow\quad x \:=\:\frac{16}{3} \:=\:5\tfrac{1}{3}$ km.