a. Explanation Let m ( t ) be the mass of the substance in the tank at time t ≥ 0 , with m ( 0 ) = m 0 . m a s s = c o n c e n t r a t i o n × v o l u m e . Assume that the mass m ( t ) is known. At some time t it changes in a small time interval [ t , t + Δ t ] to give a new mass m ( t + Δ t ) . The mass changes m ( t + Δ t ) ≈ m ( t ) + C i R Δ t − m ( t ) V R Δ t Consider the inflow pipe is opened and the sugar solution flows at the rate R is L / s . So, the volume of the solution that flows into the tank in Δ t seconds is R Δ t liters. The solution that flows into the tank has a concentration of C i g/L , therefore the mass of substance that flows into the tank in time Δ t is C i R Δ t . Now, for outflow at time t the mass of substance in the tank is m ( t ) , so the concentration of the solution is m ( t ) V g/L . With the inflow, volume of the solution that flows out of the tank in Δ t seconds is R Δ t liters and the mass of the substance that flows out of the tank in time Δ t is m ( t ) V R Δ t . This equation is an approximation because the mass of the substance changes during the time interval [ t , t + Δ t ] . The approximation improves as the length of the time interval Δ t decreases b. To determine To find: The steady-state mass of sugar in the tank. c. To determine To evaluate: The time at which the mass of sugar reach 95% of its steady level.

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 9, Problem 29RE

To determine

To find: The solution of m(t) .

To determine

To find: The steady-state mass of sugar in the tank.

To determine

To evaluate: The time at which the mass of sugar reach 95% of its steady level.

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Chapter 9, Problem 28RE

Chapter 9, Problem 30RE

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - The solution to the initial value problem y(t) = 2...

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The solution of m ( t ) .

Solution Summary: The author explains how the mass of the substance in the tank at time m(t) is known.

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To find: The solution of m(t) .

To find: The steady-state mass of sugar in the tank.

To evaluate: The time at which the mass of sugar reach 95% of its steady level.

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