# Math Help - Optimization word problem

1. ## Optimization word problem

I have a question that is as follows..

Homing pigeons avoid flying over large bodies of water, preferring to fly around them instead (One possible explanation is the fact that extra energy is required to fly over water). Assume a pigeon is released from a boat 1 mile from the shore of a lake (Point B), flies first to point P on the short and then along the straight edge of the lake to reach its home at L. If L is 2 miles from point A, the point on the shore closest to the boat, and if a pigeon needs 4/3 as much energy per mile to fly over water as over land, find the location of point P, which minimizes energy used.

So Obviously I need to find the absolute minimum, and to do this, I need to create a function, differentiate it, find the critical number and then plug that into the formula to find the extrema.

I don't know if I'm setting the function up correctly though, and I'm also having a bit of trouble differentiating the function I do have set up.

http://i1162.photobucket.com/albums/...Misc/photo.jpg

That's an image of my working, though I don't think its correct, because the final value I Get can not be correct..

2. ## Re: Optimization word problem

$E'(x)=\frac{4x}{3\sqrt{x^2+1}}-1$

3. ## Re: Optimization word problem

Originally Posted by ione
$E'(x)=\frac{4x}{3\sqrt{x^2+1}}-1$
Whoops, that's a silly mistake.

Ok, so to find the critical number is as follows....

$e '(x) = (4x / 3[x^2 + 1]^{1/2}) - 1$

$= 4x / 3(x^2 + 1)^{1/2} = 1$

$= 4x = 3(x^2+1)^{1/2}$

$=(4x/3) = (x^2 + 1)^{1/2}$

$(16x^2 / 9) = x^2 - 1$

$=16x^2 = 9x^2 - 9$

$= 7x^2 = 9$

$= x^2 = (9/7)$

$= x = (3 / (7)^{1/2})$

This seems to get an answer that is more in line with what the question is asking, but when I corrected my work in the back of the book, the critical number it came up with was this...

$3(7)^{1/2} / 7$

I can't see where I've gone wrong in the solving of the critical number though...

4. ## Re: Optimization word problem

$\frac{3}{\sqrt{7}}=\frac{3\sqrt{7}}{7}$

5. ## Re: Optimization word problem

Originally Posted by astuart
...

$=(4x/3) = (x^2 + 1)^{1/2}$

$(16x^2 / 9) = x^2 - 1$

...
See it?

Nevermind, you switched it back later (the -9 became a plus +9 for no reason). This is one of those "two wrongs do make a right" situations.

6. ## Re: Optimization word problem

Originally Posted by johnsomeone
See it?

Nevermind, you switched it back later (the -9 became a plus +9 for no reason). This is one of those "two wrongs do make a right" situations.
Whoops, didn't notice that either...Need to check my working better.