CAB-Chp-4-6B-NTs - CalcAB Chp 4-6B Related Rates, Fixed...

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CalcAB Chp 4-6B Related Rates, Fixed Amounts and Trigonometry Notes 1. Ex(right triangle, Pythagorean theorem) An airplane is flying in a flight path that will take it directly over a radar tracking station. The distance (between the plane & station) is decreasing at a rate of 400 miles per hour when the plane is 10 miles from the station. What is the speed of the plane? Organize the language the path of the plane is the horizontal (dotted line which is parallel to the ground) (not the hypotenuse, that would be a crash into the radar station) distance is decreasing at a rate of 400 mph dd/dt = 400 mph (could make negative) speed is along the path of the plane, which is the horizontal What is the speed of the plane? dx/dt = ?? mph when plane is 10 miles from station d = 10 mi. Save this value to use with the derivative later. An important aspect of each problem to look for is what actually changes and what does not. The height of the plane DOES NOT CHANGE; the plane stays on the horizontal path above the ground. The height is a FIXED amount. Because the height does not change, you have two options on how to set up the equation. Need an equation. The problem involves a right triangle and three sides , use Pythagorean theorem . Option #1: When making the equation do not use a variable for height; instead use height=6 (a leg of a right triangle). Option #2: When making the equation use a variable for height , say h, (one of the legs of a right triangle), then use dh/dt = 0 mph . dh/dt= 0 because there is no change in height with respect to time. Differentiate your equation with respect to hours (dt). When you plug in known amounts d=10 and dd/dt=400 to solve for dx/dt, you will notice you need x . Use d=10 and h=6 and Pythagorean theorem to figure out x. Answer: 6 x d
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2. Ex(volume, cylinder) From previous ASN Pg 251 #15 The mechanics at Lincoln Automotive are re-boring a six-inch deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one -thousandths of an inch every three minutes. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.8 inches? Picture is not provided with the problem. The radius that is changing is the radius of the inner core, not the radius for the whole cylinder. As in the previous example, you need to observe what parts are changing and what stay the same measure.
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This note was uploaded on 02/21/2011 for the course MATH 408C taught by Professor Knopf during the Spring '10 term at University of Texas at Austin.

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CAB-Chp-4-6B-NTs - CalcAB Chp 4-6B Related Rates, Fixed...

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