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.
Best Answer
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).