Calc04_6 - 4.6 Related Rates Greg Kelly Hanford High School...

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4.6: Related Rates Greg Kelly, Hanford High School, Richland, Washington
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First, a review problem: Consider a sphere of radius 10cm. If the radius changes 0.1cm (a very small amount) how much does the volume change? 3 4 3 V r π = 2 4 dV r dr π = ( 29 2 4 10cm 0.1cm dV π = 3 40 cm dV π = The volume would change by approximately . 3 40 cm π
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Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec. (Possible if the sphere is a soap bubble or a balloon.) 3 4 3 V r π = 2 4 dV dr r dt dt π = ( 29 2 cm 4 10cm 0.1 sec dV dt π = 3 cm 40 sec dV dt π = The sphere is growing at a rate of . 3 40 cm /sec π Note: This is an exact answer, not an approximation like we got with the differential problems.
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Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping? L 3 sec dV dt = - 3 cm 3000 sec = - Find dh dt 2 V r h π = 2 dV dh r dt dt π = ( r is a constant.) 3 2 cm 3000 sec dh r dt π - = 3 2 cm 3000 sec dh dt r π = - (We need a formula to relate V and h . )
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