Problem 8. A large tank initially contains 200L of water, into which 100kg of salt have been dissolved. At time t = 0, a valve is opened at the top that allows a solution with a salt concentration of 10kg/L to enter at a rate of 2L/min. At the same time, a valve is opened at the bottom, from which the well-mixed solution leaves the tank at a faster rate of 4L/min. a) b) c) d) First, find an equation for V(t), the volume of solution in the tank at time t. Set up an initial value problem that will allow you to find Q(t), the amount of salt (in kg) in the tank at time t. You do not need to solve it. Please refamiliarize yourself with the instructions on the first page of this exam. In particular, please explain what your variables mean and what units they are. Q(t) == 19 The solution to the IVP for this model is Q(t) =- t2+18t+100 or 100 1 100 (19t+100) (t100). (the factoring is provided for your convenience). Verify that this is a solution to the IVP you found in part (b) (again, do not directly solve the IVP). Suppose there is an emergency shut off switch when the volume of the tank reaches 5% of its original amount. When does this occur (if ever), and what is the concentration of salt in the tank at this time?
Problem 8. A large tank initially contains 200L of water, into which 100kg of salt have been dissolved. At time t = 0, a valve is opened at the top that allows a solution with a salt concentration of 10kg/L to enter at a rate of 2L/min. At the same time, a valve is opened at the bottom, from which the well-mixed solution leaves the tank at a faster rate of 4L/min. a) b) c) d) First, find an equation for V(t), the volume of solution in the tank at time t. Set up an initial value problem that will allow you to find Q(t), the amount of salt (in kg) in the tank at time t. You do not need to solve it. Please refamiliarize yourself with the instructions on the first page of this exam. In particular, please explain what your variables mean and what units they are. Q(t) == 19 The solution to the IVP for this model is Q(t) =- t2+18t+100 or 100 1 100 (19t+100) (t100). (the factoring is provided for your convenience). Verify that this is a solution to the IVP you found in part (b) (again, do not directly solve the IVP). Suppose there is an emergency shut off switch when the volume of the tank reaches 5% of its original amount. When does this occur (if ever), and what is the concentration of salt in the tank at this time?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Problem 8. A large tank initially contains 200L of water, into which 100kg of salt have been dissolved.
At time t = 0, a valve is opened at the top that allows a solution with a salt concentration of 10kg/L
to enter at a rate of 2L/min. At the same time, a valve is opened at the bottom, from which the
well-mixed solution leaves the tank at a faster rate of 4L/min.
a)
b)
c)
d)
First, find an equation for V(t), the volume of solution in the tank at time t.
Set up an initial value problem that will allow you to find Q(t), the amount
of salt (in kg) in the tank at time t. You do not need to solve it. Please refamiliarize
yourself with the instructions on the first page of this exam. In particular, please explain what
your variables mean and what units they are.
Q(t)
==
19
The solution to the IVP for this model is Q(t)
=-
t2+18t+100
or
100
1
100
(19t+100) (t100). (the factoring is provided for your convenience). Verify
that this is a solution to the IVP you found in part (b) (again, do not directly solve the IVP).
Suppose there is an emergency shut off switch when the volume of the tank reaches
5% of its original amount. When does this occur (if ever), and what is the concentration of
salt in the tank at this time?
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