Explanation Use the given initial conditions to determine the solution of the differential equation. Given: A 1000 L tank contains 500L of water with a salt concentration of 10g/L. Water with a salt concentration of 50 g/L flows into the tank at a rate of R i n = 80 L/minutes. The fluid mixes instantaneously and is pumped out at a specified rate R o u t . Let y ( t ) denote the quantity of salt in the tank at time t Calculation: Since the rate of water flow in minus the rate of water flow is 40 L / min . Therefore, at any time t , there will be 500 + 40 t liters of water in the tank. The net flow of salt into the tank will be d y d t = ( 80 ) ( 50 ) − ( 40 ) ( y 500 + 40 t ) = 4000 − 40 y 500 + 40 t This can be rewritten as d y d t + 4 y 50 + 4 t = 4000 and hence A ( t ) = 4 50 + 4 t , B ( t ) = 4000 The integrating factor is hence α ( t ) = e ln ( To determine (b) The salt concentration when the tank overflows

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Differential Equations

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Chapter 10.4, Problem 29E

To determine

(a)

Setup and solve the differential equation for $y (t)$ assuming that $R_{o u t} = 40 L / \min$

To determine

(b)

The salt concentration when the tank overflows

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Chapter 10.4, Problem 28E

Chapter 10.4, Problem 30E

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(a) Setup and solve the differential equation for y ( t ) assuming that R o u t = 40 L / min

Solution Summary: The solution of the differential equation is y ( t ) = 1, 00, 000, and 4000, respectively.

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(a)

Setup and solve the differential equation for $y (t)$ assuming that $R_{o u t} = 40 L / \min$

(b)

The salt concentration when the tank overflows

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Chapter 10 Solutions

(a) Setup and solve the differential equation for y ( t ) assuming that R o u t = 40 L / min

Solution Summary: The solution of the differential equation is y ( t ) = 1, 00, 000, and 4000, respectively.

Concept explainers

Videos

(a) Setup and solve the differential equation for y(t) assuming that Rout=40L/min

(b) The salt concentration when the tank overflows

Want to see the full answer?

Chapter 10 Solutions

(a)

Setup and solve the differential equation for $y (t)$ assuming that $R_{o u t} = 40 L / \min$

(b)

The salt concentration when the tank overflows