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# Bird Migration 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 $v \mathrm{~m} / \mathrm{s}$ is described well by the function $P(v)=17 v^{-1}+10^{-3} v^3 \mathrm{~J} / \mathrm{s}$. Assume that the pigeon can store $5 \times 10^4 \mathrm{~J}$ of usable energy as body fat. (a) Find the velocity $v_{\text {pmin }}$ that minimizes power consumption.

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Note that the birds will exhaust their energy after

$T= \dfrac{5\cdot10^4 \text{ joules}}{P(v) \text{ joules/sec}}=\dfrac{5\cdot10^4}{P(v)} \text{ sec}$

Then, the total distance traveled after using their stored energy is

$D(v)=vT=\dfrac{(5\cdot10^4)v}{P(v)}$

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