Explanation Given: Volume and outflow in Great Lakes: V (km 3 ) R (km 3 /year) Superior 12,200 65.2 Michigan 4900 158 Erie 460 175 Ontario 1600 209 Calculation: The differential equation for Q changing with time, Rate of Q changes = − ( Rate of pollutants leave in outflow ) where the negative sign represents the fact that Q is decreasing. At time t, the concentration of pollutants is Q / V and water containing this concentration is leaving at rate r. Thus, Rate pollutants leave in outflow = Rate of outflow × Concentration = r ⋅ Q V . So, the differential equation is d Q d t = − r V Q and its general solution is Q = Q 0 e − r t / V For Lake Superior, r V = 65.2 12200 = 0.005 , so, at time t we have, Q = Q 0 e − 0.005 t . When 80% of the pollution has been removed, 20% remains, so Q = 0.2 Q 0 . Substituting gives 0.2 Q 0 = Q 0 e − 0.005 t Cancelling Q 0 and solving for t gives ln ( 0.2 ) = − 0.005 t t = − ln ( 0.2 ) 0.005 t ≈ 322 years For Lake Michigan, r V = 158 4900 = 0.032 , so, at time t we have Q = Q 0 e − 0.032 t . When 80% of the pollution has been removed, 20% remains, so Q = 0.2 Q 0 . Substituting gives 0.2 Q 0 = Q 0 e − 0.032 t Cancelling Q 0 and solving for t gives ln ( 0

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Chapter 9.4, Problem 12P

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The Great Lake which would require the longest time and the shortest time for 80% of the pollution to be removed and also calculate the ratio of these two times.

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The Great Lake which would require the longest time and the shortest time for 80% of the pollution to be removed and also calculate the ratio of these two times.

Solution Summary: The author calculates the differential equation for Q changing with time, where the negative sign represents the fact that Q is decreasing.

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To find: The Great Lake which would require the longest time and the shortest time for 80% of the pollution to be removed and also calculate the ratio of these two times.

Chapter 9 Solutions

To find:

The Great Lake which would require the longest time and the shortest time for 80% of the pollution to be removed and also calculate the ratio of these two times.