Question

2370315 [0/4 Points) DETAILS PREVIOUS ANSWERS SCALC8 3.7.079. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 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 6 km from the nearest point 8 on the shoreline, flies to a point 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 11 km apart. (Round your answers to two decimal places.) (a) In general, if it takes 1.4 times as much energy to fly over water as land, to what point should the bird fly in order to minimize the total energy expended in returning to its nesting area? X km from B 16.67 (b) Let Wand 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. w 36 + x2 L x х (c) What should the value of W/L be in order for the bird to fly directly to its nesting area D? 2.068 x (d) If the ornithologists observe that birds of a certain species reach the shore at a point 2 km from B, how many times more energy does it take a bird to fly over water than land? 1.11

Answer 1

Solution - The detailed solution is given below, please find the attached pages -

page no 1

island Solution : (a) x²+ 36 6 km D nest B K Il km Ex-st ll-x - Hence if K= energy Ikm over land, then energy / km over water = 1.4 K Hence the total energy is - E- 1.4K VX2+36 + K (11-X) 0SX S11 sol 1.4 K x DE da 11 ( 36+22) 2 Set de da Henu, 1.4ka K VX2+36 1.44= VX? +36 1-96x?- X2+36 0.96x? - 36 x= 36 0.96 = 37.5 x= 6.12 km o

page no 2

at the endpoints. Testing against the value of E E(0) = 1.4K (6) + LIK E (0) = 19.4k E(5-3) = |-4 Rx 789 + kỷ 5.9 E(6.12 = 17-89 K. and E(1)– 1.4XK 157 + (-2) K. El.) - IS.54 K Thus, to minimize energy, the birds should fly to a point about 6.12 km from 8. Are (b) If W/L is large, the bird should fly to a point c that is closer to B than to D to minimize the energy used over water. If W/L is small, the bird would fly to a point c that is closer to D than to B to minimize the distance of the flight E = WV X2+36 + 1111-x) Wa -L de da x²+36 for maxima f minima, dE da (2

page no 4

W2 = L √x²+36 de ♡ VX?+36 By the same sort of argument as in part (a), this ratio will give the minimal expenditure of energy the bird heads for the point x km from B. for flight direct to D, X=11 so from equation (1), W 112+36 if ( C:) 12:52 11 11 1-139 There is no value of W/L for which the bird should fly directly to B. But note that lim. (WL) = c, so if the point at 2+0+ close to B, then W/L Which E is a minimum is is large. 3

page no 4

(d) Ansuming that the birds sites instinctively choose the path that minimizes the energy expenditure, we can use the equation for de co da from part (a) with- 1-4 K - C 2-2 and KPI (9)= 10V 367 4. v 40 - C 2 C= 3.16 km 4)

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