The Joy of Flying. It takes a certain power to keep a plane moving along at a sp

Question

The Joy of Flying. It takes a certain power to keep a plane moving along at a speed . The power needs to overcome air drag which increases as the speed increases, and it needs to keep the plane in the air which gets harder as the speed decreases. So assume the power required is given by P=cv^2+(d/v^2) where c and d are positive constants. (They depend on your plane, your altitude, and the weather, among other things.) Enter here [ ? ] the choice of v that will minimize the power required to keep moving at speed v . Suppose you have a certain amount of fuel and the fuel flow required to deliver a certain power is proportional to to that power. What is the speed v that will maximize your range (i.e., the distance you can travel at that speed before your fuel runs out)? Enter your speed here [ ? ] Finally, enter here [ ? ] the ratio of the speed that maximizes the distance and the speed that minimizes the required power.

Explanation / Answer

rate me first P = cv^2 + d/v^2 = cv^2 + dv^-2 A. We want minimum P Find the derivative dP/dv and set it = 0 dP/dv = 2cv - 2dv^-3 = 0 2cv = 2dv^-3 v^4 = d/c v(min-fuel) = (d/c)^(1/4) B. For maximum range we want minimum P/v Find the derivative d(P/v)/dv and set it = 0 P/v = cv + dv^-3 d(P/v)/dv = c - 3dv^-4 = 0 c = 3dv^-4 v^4 = 3d/c v(max-range) = (3d/c)^(1/4) C. v(max-range)/v(min-fuel) = (3d/c)^(1/4) / (d/c)^(1/4) = 3^(1/4) = 1.316074

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