Explanation Given information: The rate of change of the volume of water in a vase against the time follows the function mentioned below. V ' ( t ) = − 0.05 π ( 10 − 0.2 t ) 2 per hour Because of the evaporation the volume of water is decreasing with time and the vase contains 200 c m 3 of water. Formula used: The net change theorem is used. ∫ a b V ' ( t ) d t = V ( b ) − V ( a ) Calculation: According to the net change theorem, the integral of the rate of change of the volume returns the volume at time t . Hence, ∫ V ' ( t ) d t is the existing volume of water at time t . An assumption is made as the time it will take to empty the vase by evaporation is T . Then, when t = 0 , the volume of water is 200 c m 3 and when t = T the volume is zero. Applying the net change theorem to the above scenario, ∫ 0 T V ' ( t ) d t = V ( T ) − V ( 0 ) Here, V ( T ) = 0 and V ( 0 ) = 200 . ∫ 0 T V ' ( t ) d t = V ( T ) − V ( 0 ) ∫ 0 T − 0.05 π ( 10 − 0.2 t ) 2 d t = 0 − 200 − 0

5th Edition

ISBN: 9781429241861

Author: Laura Taalman, Peter Kohn

Publisher: W. H. Freeman

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Chapter 4.5, Problem 68A

To determine

To find: The time it will take to empty all the water in the vase by the evaporation using the Net Change Theorem.

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Chapter 4.5, Problem 67A

Chapter 4.5, Problem 69A

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The time it will take to empty all the water in the vase by the evaporation using the Net Change Theorem.

Solution Summary: The author explains how the net change theorem calculates the time it takes to empty the vase by evaporation.

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