Two cylindrical water tanks are vertically connected to each other. Initially, the top and bottom tank contain 100 litres and 200 litres of water, respectively. At time t = 0, the valve between the tanks is opened. The flow rate through each of these valves is proportional to the volume of the water in the tank above the valve. The rate at which the volume for both tanks are given in the following differential equations. dvi = Vy dt Rate of volume change for the top tank, Rate of volume change for the bottom tank, dvz dt a) By taking Llv, ()) = V,(6) and L[vz(t)] = V,(6), show that %3D 100s 100 200 V,(s) = 5 +1 200s V2(s) = +1 s +1 and %3D s +1 b) Use Laplace transforms to determine v, (t) and v2 (t). Then, find the volume of the water in the top and bottom tank after 30 minutes.
Two cylindrical water tanks are vertically connected to each other. Initially, the top and bottom tank contain 100 litres and 200 litres of water, respectively. At time t = 0, the valve between the tanks is opened. The flow rate through each of these valves is proportional to the volume of the water in the tank above the valve. The rate at which the volume for both tanks are given in the following differential equations. dvi = Vy dt Rate of volume change for the top tank, Rate of volume change for the bottom tank, dvz dt a) By taking Llv, ()) = V,(6) and L[vz(t)] = V,(6), show that %3D 100s 100 200 V,(s) = 5 +1 200s V2(s) = +1 s +1 and %3D s +1 b) Use Laplace transforms to determine v, (t) and v2 (t). Then, find the volume of the water in the top and bottom tank after 30 minutes.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Two cylindrical water tanks are vertically connected to each other. Initially, the top and
bottom tank contain 100 litres and 200 litres of water, respectively. At time t= 0, the valve
between the tanks is opened. The flow rate through each of these valves is proportional to
the volume of the water in the tank above the valve. The rate at which the volume for both
tanks are given in the following differential equations.
dvi
= V,
dt
Rate of volume change for the top tank,
Rate of volume change for the bottom tank, dvz
dt
= -V
a) By taking Llv, ()) = V,(6) and L[vz(t)] = V,(6), show that
%3D
100s
100
200
V,(s) = 5 +1
200s
V,(s) = 52 +1
and
s? + 1
g2 +1
b)
Use Laplace transforms to determine v, (t) and v2 (t). Then, find the volume of the
water in the top and bottom tank after 30 minutes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d61f17a-9ba2-4e5d-9d30-80516a3dc340%2F8c7dff4c-ac88-4d14-b861-3b1439be75a8%2Fkepycb8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Two cylindrical water tanks are vertically connected to each other. Initially, the top and
bottom tank contain 100 litres and 200 litres of water, respectively. At time t= 0, the valve
between the tanks is opened. The flow rate through each of these valves is proportional to
the volume of the water in the tank above the valve. The rate at which the volume for both
tanks are given in the following differential equations.
dvi
= V,
dt
Rate of volume change for the top tank,
Rate of volume change for the bottom tank, dvz
dt
= -V
a) By taking Llv, ()) = V,(6) and L[vz(t)] = V,(6), show that
%3D
100s
100
200
V,(s) = 5 +1
200s
V,(s) = 52 +1
and
s? + 1
g2 +1
b)
Use Laplace transforms to determine v, (t) and v2 (t). Then, find the volume of the
water in the top and bottom tank after 30 minutes.
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