Two cylindrical water tanks are vertically conected to each otner. ly, the 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. dv1 = v2 dt Rate of volume change for the top tank, Rate of volume change for the bottom tank, dv2 =ーV1 dt a) By taking L[v (t)] = V, (s) and L[v2(t)] = V2 (s), show that %3D 200 100s 200s 100 V, (s) = and V2 (s) = s2 +1 s2 +1 s2 + 1 s2 +1 b) Use Laplace transforms to determine v, (t) and v, (t). Then, find the volume of the water in the top and bottom tank after 30 minutes.
Two cylindrical water tanks are vertically conected to each otner. ly, the 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. dv1 = v2 dt Rate of volume change for the top tank, Rate of volume change for the bottom tank, dv2 =ーV1 dt a) By taking L[v (t)] = V, (s) and L[v2(t)] = V2 (s), show that %3D 200 100s 200s 100 V, (s) = and V2 (s) = s2 +1 s2 +1 s2 + 1 s2 +1 b) Use Laplace transforms to determine v, (t) and v, (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
Rate of volume change for the top tank,
= v2
dt
Rate of volume change for the bottom tank, dvz
= -v1
dt
a)
By taking L[v, (t)] = V, (s) and L[v2(t)] = V2(s), show that
200
100s
200s
100
V1 (s)
and
V2 (s):
%3D
s2 + 1
s2 + 1
s2 +1
s2 + 1
b)
Use Laplace transforms to determine v, (t) and v, (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%2F67559312-114c-47b6-accc-19eb181fdd96%2F691efc7c-f65e-4ba7-ba72-9ce6709aa890%2Fyrhin5v_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
Rate of volume change for the top tank,
= v2
dt
Rate of volume change for the bottom tank, dvz
= -v1
dt
a)
By taking L[v, (t)] = V, (s) and L[v2(t)] = V2(s), show that
200
100s
200s
100
V1 (s)
and
V2 (s):
%3D
s2 + 1
s2 + 1
s2 +1
s2 + 1
b)
Use Laplace transforms to determine v, (t) and v, (t). Then, find the volume of the
water in the top and bottom tank after 30 minutes.
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