Minimum distance/energy bird flight problem.

[kieth89]

Minimum distance/energy bird flight problem. (SOLVED)

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

Homework Statement

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.

Homework Equations

Energy used = Energy per mile * miles flown

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).

 

[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.

 

[gopher_p]

$14^2=196$

 

[kieth89]

$14^2=196$

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.

 

[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 canceled and not had to worry about it in the first place.

Sometimes not simplifying in one step makes it easier to simplify in the next. But sometimes it works the other way around. You just got to 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.