408DS09assign9

# 408DS09assign9 - a and b(measured in cm and both sides are...

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M408D Spring 2009 Assignment 9 Due Thursday, April 16 Be sure that you have read and understood sections 15.4, 15.5, and 15.6 and worked the assigned text exercises before you complete this assignment. 1. Consider the surface deﬁned by the equation xyz = 27, x > 0, y > 0. (a) Find the equation for the tangent plane to the surface at a point x 0 ,y 0 , 27 x 0 y 0 · . (b) Show that all pyramids formed by the coordinate planes and a tangent plane to the surface xyz = 27 have the same volume. (The volume of a pyramid is 1 3 Bh where B is the area of the base and h is the height of the pyramid.) 2. A petrochemical company uses a cylindrical tank with hemispherical ends to transport its product. Such a tank has radius 3 feet and the height of the cylindrical portion is 15 feet. A coat of paint 1/10th of an inch is applied to the entire tank. Use diﬀerentials to approximate the amount of paint needed. 3. Two sides of a triangle have lengths
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Unformatted text preview: a and b (measured in cm) and both sides are increasing at the rate of 1 cm/sec. If the area of the triangle remains constant, at what rate is the angle θ between a and b changing when θ = π 6 , a = 4, and b = 3? 4. The temperature at each point of a metal plate is given by the function T ( x,y ) = 100 e-x 2-3 y 2 , where T is measured in degrees Celsius and x and y are measured in meters. (a) If an object at the point (3, -1) moves toward the point (1,2), does the temperature increase or decrease? At what rate? (b) Starting from the point (3, -1), in what direction does the temperature increase most rapidly? What is the rate of change in this direction? (c) Starting from the point (3, -1), in what direction does the temperature decrease most rapidly? What is the rate of change in this direction? 1...
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## This note was uploaded on 10/14/2009 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

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