# Minimum distance/energy bird flight problem.

1. May 14, 2012

### kieth89

Minimum distance/energy bird flight problem. (SOLVED)

SOLVED: Problem was just a simple arithmetic error (14 squared)

1. The problem statement, all variables and given/known data
A bird is released from point A on an island 5 mi from the nearest point B on a straight shoreline. The bird flies to a point C on the shoreline and then flies along the shoreline to its nesting area D. Suppose the bird requires 10 kcal/mi of energy to fly over land and 14 kcal/mi to fly over water. The distance between point B and D is 12 miles.

View the attached diagram for an image.

If the bird instinctively chooses a path that minimizes its energy expenditure, what is its path?
Note: We are supposed to solve this using limit and derivative.
2. Relevant equations
Energy used = Energy per mile * miles flown

3. The attempt at a solution
I termed the distance from point B to C $x$, and the distance from point C to D $12 - x$.
I then found the distance from point A to point C to be $\sqrt{25 + x^{2}}$. Using that I found the equation for the total energy used to be as follows:
$Total Energy Used = 14\sqrt{25+x^{2}} + 10(12 - x)$ This formula is the distance flown over water (Point A to C) + the distance flown over land (C to D), both of those times their energy used per mile.

I then found the derivative of that equation (zooming in (Ctrl + usually) with the browser will make the math easier to read):
$f^{1}(x) = \lim_{h \to 0} \frac{14\sqrt{25+(x+h)^{2}} + 10(12 - x - h) - 14\sqrt{25 + x^{2}} - 120 + 10x}{h}$
$f^{1}(x) = \lim_{h \to 0} \frac{14\sqrt{25+(x+h)^{2}} + 120 - 10x - 10h - 14\sqrt{25 + x^{2}} - 120 + 10x}{h}$
$f^{1}(x) = \lim_{h \to 0} \frac{14\sqrt{25+(x+h)^{2}} - 10h - 14\sqrt{25 + x^{2}}}{h}$
$f^{1}(x) = \lim_{h \to 0} \frac{14\sqrt{25+(x+h)^{2}} - 14\sqrt{25 + x^{2}}}{h} - \lim_{h \to 0} \frac{10h}{h}$
$f^{1}(x) = \lim_{h \to 0} (\frac{14\sqrt{25+(x+h)^{2}} - 14\sqrt{25 + x^{2}}}{h}) (\frac{14\sqrt{25+(x+h)^{2}} + 14\sqrt{25 + x^{2}}}{14\sqrt{25+(x+h)^{2}} + 14\sqrt{25 + x^{2}}}) - \lim_{h \to 0} 10$
$f^{1}(x) = \lim_{h \to 0} \frac{186(25 + x^{2} + 2xh + h^{2}) - 186(25 + x^{2})}{14h\sqrt{25 + (x+h)^{2}} + 14h\sqrt{25 + x^{2}}} - \lim_{h \to 0} 10$
$f^{1}(x) = \lim_{h \to 0} \frac{4650 + 186x^{2} + 372xh + 186h^{2} - 4650 - 186x^{2}}{14h(\sqrt{25 + (x+h)^{2}} + \sqrt{25 + x^{2}})} - \lim_{h \to 0} 10$
$f^{1}(x) = \lim_{h \to 0} \frac{372xh + 186h^{2}}{14h(\sqrt{25 + (x+h)^{2}} + \sqrt{25 + x^{2}})} - \lim_{h \to 0} 10$
$f^{1}(x) = \lim_{h \to 0} \frac{372x + 186h}{14(\sqrt{25 + (x+h)^{2}} + \sqrt{25 + x^{2}})} - \lim_{h \to 0} 10$
$f^{1}(x) = \lim_{h \to 0} \frac{372x + 186(0)}{14\sqrt{25 + (x+(0))^{2}} + 14\sqrt{25 + x^{2}}} - \lim_{h \to 0} 10$

Simplifying derivative:
$f^{1}(x) = \frac{372x}{14\sqrt{25 + x^{2}} + 14\sqrt{25 + x^{2}}} - 10$
$f^{1}(x) = \frac{372x}{28\sqrt{25 + x^{2}}} - 10$

Set derivative equal to 0 and solve:
$0 = \frac{372x}{28\sqrt{25 + x^{2}}} - 10$
$0 = \frac{372x - 280\sqrt{25 + x^{2}}}{28\sqrt{25 + x^{2}}}$
$0 = 372x - 280\sqrt{25 + x^{2}}$
$-372x = - 280\sqrt{25 + x^{2}}$
$\frac{372}{280}x = \sqrt{25 + x^{2}}$
$\frac{138384}{78400}x^{2} = 25 + x^{2}$
$\frac{8649}{4900}x^{2} = 25 + x^{2}$
$0 = 25 + x^{2} - \frac{8649}{4900}x^{2}$
$0 = 25 - \frac{3749}{4900}x^{2}$
$0 = (5 - \frac{\sqrt{3749}}{70}x)(5 + \frac{\sqrt{3749}}{70}x)$
$0 = 5 - \frac{\sqrt{3749}}{70}x$ and $0 = 5 + \frac{\sqrt{3749}}{70}x$
Since x is distance, we know it can't be negative, therefore the (5 + fraction x = 0) is not a solution, which leaves the first one.
$0 = 5 - \frac{\sqrt{3749}}{70}$
$-5 = -\frac{\sqrt{3749}}{70}$
$-5 * - \frac{70}{\sqrt{3749}}= x$
$x = 5.716238$
Which means that the distances are as follows:
The bird flies around 7.59 miles over water and around 6.28 miles over land..

However, the book says the solution for x is around 5.1 miles.

So, where did I go wrong, or did not rounding skew with my answer? The book's answer does yield around a .17 less energy use compared to mine when plugged into the original formula. But that is a very small difference. Thanks for any help.
Oh, and also, did I do my notations right with limits (ignoring LaTex putting the h to 0 beside the lim sign).

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Last edited: May 14, 2012
2. May 14, 2012

### gopher_p

Why are you using the definition of the derivative? Why not power rule?

Edit: Sorry. Guess I didn't read all of your instructions.

3. May 14, 2012

### gopher_p

$14^2=196$

4. May 14, 2012

### kieth89

Figured it'd be something like that..Didn't have a calculator on hand when doing this, and hadn't done long multiplication by hand for a looong time. Thanks for the help.

5. May 14, 2012

### gopher_p

Understandable. Though if you had just left it as $14^2$, you may have recognized that all of the terms in that rational expression had a 14 ... so you could have cancelled and not had to worry about it in the first place.:tongue2:

Sometimes not simplifying in one step makes it easier to simplify in the next. But sometimes it works the other way around. You just gotta practice to see when. I always suggest people use calculators as little as possible when time isn't a factor. There are a lot of benefits to keeping your arithmetic skills strong.

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