2007-10-05 Example 1 - A rising balloon

Example - -~~ = 0.14 rad/min when e = n/4 dy Y dt when e = 7/4 500 ft Step 2 Write down the additional numerical informarion da-= 0.14rad/min dl

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EXAMPLE i A rising balloon A hot-air balloon rising straight up from a level field is tracked by a range finder 500 ft from the lift-off point. At the moment the range finder's elevation angle is )(/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon rising at that moment? Solution We answer the question in six steps. Step 1: Draw a picture and name the variables and constants. variables in the picture are e = the angle the range finder makes with the ground (radians) y = the height of the ballo.on(feet). We let t represent time and assume a and y to be differentiable functions of t. The one constant in the picture .is the distance from the range finder to the lift-off point (500 ft). There is no need to give it a special symbol. The Rangefinder / Balloon
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Unformatted text preview: -~~ = 0.14 rad/min when e = n/4 dy _? Y'/ dt- . when e = 7[/4 500 ft Step 2: Write down the additional numerical informarion. da .-= 0.14rad/min dl when Step 3: Write down what we are asked to find. We wane dy /dl when a = 7T /4. Step 4: Write an equation that relates the variablesy and e. y-= tan a, or y = 500tan a 500 Step 5: Differentiate with' respect to t using the Chain Rule. The result tells how dy/dt (which we want) is related to de/de (which we know). dy ? de .-= 500sec-e-dt dt Step 6: Evaluate with a = )(/4 and da/dt = 0.14 to find dy/dt. d . . .~ = 500(,J2)2(0.14) = (1000)(0.14) = 140 dt . see .:::.=.J2 4 At .themoment in question, the balloon is rising at the rate of 140 ft/rnin....
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This note was uploaded on 09/08/2008 for the course EAC 101 taught by Professor Tyler/ralston during the Fall '08 term at University of Louisville.

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