Question

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 5 km from the nearest point B on a straight 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 13 km apart.

(1 point) is believed that more energy is required Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies the nearest point B on a straight 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 13 km apart. released from an island that is 5 km from (a) In general, if it takes 1.4 times as much energy to fly over the water as land, how far should point C be from B in order to minimize the total energy expended in returning to its nesting area? .103 km (b) Let W and L denote the energy (in joules) per kilometer flown over water and land respectively. Determine the ratio W/L corresponding to the minimum energy expenditure given that the bird flies to the shore at a point x kilometers from B. (Your answer may depend only on x.) W/L= (c) If the ornitholog ists observe that birds of a certain species reach the shore at a point 1 km from B, how many times more energy does it take this species to fly over water than land? island 5 km nest 13 km

(a) In general, if it takes 1.4 times as much energy to fly over the water as land, how far should point C be from B in order to minimize the total energy expended in returning to its nesting area?

5.103 km

(b) Let W and L denote the energy (in joules) per kilometer flown over water and land respectively. Determine the ratio W/L corresponding to the minimum energy expenditure given that the bird flies to the shore at a point x kilometers from B. (Your answer may depend only on x.)

W/L = _____

(c) If the ornithologists observe that birds of a certain species reach the shore at a point 1 km from B, how many times more energy does it take this species to fly over water than land?

My answer for (a) is correct, but I'm struggling with (b) and (c).

Answer 1

Auwes A Islaud km Net t km Jt xthe dis hance om 8doC (ay The distan ce Hom the island toC is and the ditance tomc to Dis (3-) The total olis tan ce to havel is z2r +03-x) J ke the eneugy ber km Jt takato over daud thay the to tal euergy s E - 1.4k 225 k(13-x) , 1.4kx k dE 1.4kx -k o LoTh sides 1.kz -k 1-96x2-72+ 2s 1.9622-x2s 0.9671 25 2 A S. O.94 D-96 estng agant the value ot E at he eboists r)+13k 2ok E C12) 19-5k, Thu, to minimize E (S.1) K about 5. km from sngy, the nd shold port 13.

fehut C Tht laige ,The bd oould claser to B Thau to D to minimize the over The water. I w/L small, te ind wold poit C that chser to D thau B do niwie The dis tance t The Hipht to eugy ued to a E: Wx + 13-x +25 x+5 2 2. 2 WIL C) 26 - 5.0990/9s) 5 times