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3.8.4-eg-1 - Example A spotlight is on the ground 20 ft...

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Example A spotlight is on the ground 20 ft away from a wall and a 6 ft tall person is walking toward the wall at a rate of 5 2 ft/sec. How fast is the height of the shadow changing when the person is 8 ft from the wall? Solution: The variables marked in the following picture , are: x ( t ) = changing distance of the person from the spotlight at time t , z ( t ) = changing distance of the person from the wall at time t ( z ( t ) = 20 - x ( t )), and y ( t ) = changing height of the shadow on the wall. We are given that z 0 ( t ) = - 5 2 (negative because the distance function z ( t ) is decreasing with time); we are also given that at a particular instant t ? in time, z ( t ? ) = 8 , so x ( t ? ) = 20 - 8 = 12 . The problem is then asking for the rate of change of the height of the shadow at the instant t ? ; that is, we want y 0 ( t ? ) . To determine properties about y 0 ( t ) from the known rate z 0 ( t ), it is necessary to relate the quantities y ( t ) and z ( t ). This can be done by noting that there are two similar right triangles in the picture given above:
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The larger triangle has vertices at the spotlight, the base of the wall, and at the upper tip of the shadow on the wall; this triangle has sides of length 20, y ( t ), with hypotenuse
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