# Math Help - Optimization

1. ## Optimization

Ornithologists have determined that 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 generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 4 km from the nearest point B on the shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 12 km apart. Give your answers to two decimal places.

(a) In general, if it takes 1.3 times as much energy to fly over water as land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area?

(b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. Assuming the bird's energy expenditure is minimized, determine a function for the ratio W/L in terms of x, the distance from B to C.

I'm assuming (b) is reliant on (a), so here's my work for (a):

(the line) BC = L
(the line) AC = w = $\sqrt{L^2+16}$

e = 1.3w + (12-L)
e = 1.3 $\sqrt{L^2+16}$ + 12 - L
$e' = \frac{1.3*L}{\sqrt{L^2+16}}-1$

Which gave me a minimum of like L = 6.08, which isn't correct. Where did go wrong?

2. The problem says that the rate over water is 1.3 times as much as over land, so try dividing the distance on water over this rate instead of multiplying it and see if you get the right answer.