Question

Power P is the rate at which energy E is consumed per unit time. Ornithologists have found that the power consumed by a certain pigeon flying at velocity v m/s is described well by the function P(v)=16v−1+10−3v3 J/s. Assume that the pigeon can store 5×104 J of usable energy as body fat. Find the velocity vp min that minimizes power consumption. (Use decimal notation. Give your answer to two decimal places.)

Answer 1

Step-by-step explanation:

P(v) = 16 / v + 10⁻³ v³

differentiating on both sides

dP / dt = - 16 / v² + 3 x 10⁻³ v²

For maxima and minima , the condition is

dP / dt = - 16 / v² + 3 x 10⁻³ v² = 0

v² = 160 / 3 x 10²

v² = 73 m/s

v = 8.54 m /s

To know the condition of minima

again differentiating

d²P / dt² = - 16 x -2 / v² + 6 x 10⁻³ x v

= 32 / v³ + 6 x 10⁻³ x v

= + ve quantity

So at v_p = 8.54 m /s , power consumption will be minimum .