This is a multiple choice question in one of tests I just wrote and I did not know the answer to it. I was just stuck on this during the test. It is a very weird question, one I find to be impossible.

Here it goes:

Scientists have determined some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air rises over land and falls over water during day. A bird with these tendencies is released from an island that is 5 km form the nearest point B on a straight shoreline, flies to point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the brid instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart.

If it takes 2 times as much energy to fly over water as land, what is the minimum amount of total energy that the bird could expend returning to its nesting area?

a) 2.88 units $\phantom{abcd}$ b) 21.66 units $\phantom{abcd}$ c) 23.00 units $\phantom{abcd}$ d) 27.85 units

I have no idea how to do this question because it makes absolutely no sense to me. I have sat and thought but I just am not getting anywhere.

  • 1
    $\begingroup$ You must be leaving something out. The question is unanswerable as it stands. What is a unit of energy? $\endgroup$ – TonyK Nov 30 '11 at 13:03
  • $\begingroup$ The answer is b). I have to agree with TonyK that the question is pretty badly formulated. Did they at least put a drawing next to it? $\endgroup$ – Raskolnikov Nov 30 '11 at 13:14
  • $\begingroup$ That is exactly what I am thinking....energy is joules what the heck are they referring to :S $\endgroup$ – Raynos Nov 30 '11 at 13:39


This is a minimization problem, and you have to minimize

$$E(x) = 2\cdot |AC| + |CD|$$

where $x = |BC|$, with $C \in[B,D]$, $A\hat{B}D=90°$, $|AB|=5$ and $|BD|=13$.

That is,

$$E(x) = 2 \cdot \sqrt{5^2+x^2}+13-x$$

Take the derivative of this and find the zeros in $[0, 13]$, etc., giving the multiple choice answer (b).

  • 1
    $\begingroup$ Heh, it's not to scale, but I wanted to figure out how to make & post a drawing. ;))) $\endgroup$ – r.e.s. Nov 30 '11 at 14:21
  • $\begingroup$ how do u guys add the math symbols and drawing on this forum? $\endgroup$ – Raynos Nov 30 '11 at 15:34
  • $\begingroup$ thanks so much if possible can you please explain how u got the 2* |AC| + |CD|? $\endgroup$ – Raynos Nov 30 '11 at 15:38
  • $\begingroup$ @Nadal: See Markdown Editing Help. For images, there's a "picture icon" in edit mode that lets you link to or upload an image. $\endgroup$ – r.e.s. Nov 30 '11 at 15:41
  • $\begingroup$ @Nadal this is because travel over sea ($|AC|$) is twice as expensive over land($|CD|$) $\endgroup$ – ratchet freak Nov 30 '11 at 15:48

Let $x$ be the distance from where the bird starts to $C$. Using the Pythagorean Theorem, one finds that the distance from $C$ to $D$ is $13-\sqrt{x^2-25}$ . Setting the energy expenditure factor over the shore equal to $a\,{\rm units\over km}$, the energy used is $$ E(x)=2ax + a \cdot(13-\sqrt{x^2-25}). $$

What you need to do is find the minimum value of $E$ over the interval $[5,\sqrt{194}]$ (if the bird first flies to $B$, then $x=5$ ; if the bird flies straight to $D$, then $x=\sqrt{194}$).


Set $E'(x)=0$ and find any solutions in $[5,\sqrt{194}]$.

Evaluate $E$ at the solutions found above, at $x=5$, and at $x=\sqrt{194}$.

Select the smallest of these numbers. This will give you the minimum energy expenditure if the energy expenditure over land is $a\,{\rm units\over km}$.

By the way, the problem seems ill-posed, since they did not tell you what the energy expenditure over land was (I presume they took $a=1$).

  • $\begingroup$ er the shore $[BD]$ is straight and $C$ is a point on that shore so $|CD|=|BD|-|CB|=13-x$ with $|AC|^2=|AB|^2+|CB|^2=25+x^2$ $\endgroup$ – ratchet freak Nov 30 '11 at 13:25
  • $\begingroup$ I made exactly the same mistake the first time through! 13 is the distance from $B$ to $D$, not the distance from the island to $D$. So your twelves should be thirteens, and your thirteeens should be $\sqrt {194}$s. $\endgroup$ – TonyK Nov 30 '11 at 13:27

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