HH_14_6 - x pounds of cattle t weeks after the cattle...

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(11/11/08) Math 10C. Lecture Examples. Section 14.6. The Chain Rule Example 1 A small plane uses gasoline at the rate of r = r ( h , v ) gallons per hour when it is Fying at an elevation of h feet above the ground and its air speed is v knots (nautical miles per hour). At a moment when the plane has an altitude of 8000 feet and a speed of 120 knots, its height is increasing 500 feet per minute and it is accelerating 3 knots per minute. At what rate is its gasoline consumption increasing or decreasing at that moment if at h = 8000 and v = 120 , r h = - 2 × 10 - 4 gallons per hour per foot, and r v = 0 . 13 gallons per hour per knot? (1) Answer: The plane’s rate of fuel consumption is increasing 0.29 gallons per hour per minute. Example 2 A farmer will earn P = P ( x , t ) dollars pro±t if he sells
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Unformatted text preview: x pounds of cattle t weeks after the cattle market opens for the season. The total weight of his herd is a function x = x ( t ) of the time. Express the rate of change of his prot with respect to time in terms of the derivatives of P ( x , t ) and x ( t ) . Answer: d dt [ P ( x ( t ) , t )] = b P x ( x ( t ) , t ) dollars pound Bb x ( t ) pounds week B + b P t ( x ( t ) , t ) dollars week Bb d dt ( t ) weeks week B = P x ( x ( t ) , t ) x ( t ) + P t ( x ( t ) , t ) dollars per week Lecture notes to accompany Section 14.6 of Calculus by Hughes-Hallett et al. (1) Data adapted from Cessna 172N Information Manual , Wichita Kansas: Cessna Aircraft Company, 1978, p.5-16. 1...
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