Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at the bottom. The change in volume can be modelled by considering the amount of water entering and leaving the tank (per unit of time) as follows fin - fout. Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50. dV dt (a) Assume the rainfall throughout the day is getting heavier, such that, fin (t) = 10 + t and that the tank is losing 10% of its volume of water per hour, such that, fout (V)= 0.1V. Check by direct substitution that the function (b) with time: V(t) = 10(t+5e-0.1) satisfies the ODE and the initial condition. Now assume that the hole is slowly growing in size, so that the flow rate out increases fout (V, t) = 0.1(1+0.1t)V. Solve the ordinary differential equation for V(t) with this flow ra e out, using either separation of variables, or the integrating factor method. Make sure to also include the initial condition.
Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at the bottom. The change in volume can be modelled by considering the amount of water entering and leaving the tank (per unit of time) as follows fin - fout. Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50. dV dt (a) Assume the rainfall throughout the day is getting heavier, such that, fin (t) = 10 + t and that the tank is losing 10% of its volume of water per hour, such that, fout (V)= 0.1V. Check by direct substitution that the function (b) with time: V(t) = 10(t+5e-0.1) satisfies the ODE and the initial condition. Now assume that the hole is slowly growing in size, so that the flow rate out increases fout (V, t) = 0.1(1+0.1t)V. Solve the ordinary differential equation for V(t) with this flow ra e out, using either separation of variables, or the integrating factor method. Make sure to also include the initial condition.
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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Transcribed Image Text:Consider a water tank that is being filled with rainfall at the top and is drained through a small hole at
the bottom. The change in volume can be modelled by considering the amount of water entering and
leaving the tank (per unit of time) as follows
fin - fout.
Here volume of water in the tank V is in litres, flow rate in fin is in litres per hour, flow rate out fout is
in litres per hour and time t is in hours. Initially, the tank contains 50L of water: V(0) = 50.
dV
dt
(a)
Assume the rainfall throughout the day is getting heavier, such that, fin(t) = 10+t and
that the tank is losing 10% of its volume of water per hour, such that, fout (V) = 0.1V. Check by
direct substitution that the function
(b)
with time:
V(t) = 10(t + 5e-0.1t)
satisfies the ODE and the initial condition.
Now assume that the hole is slowly growing in size, so that the flow rate out increases
fout (V, t) = 0.1(1+0.1t)V.
Solve the ordinary differential equation for V(t) with this flow ra e out, using either separation of
variables, or the integrating factor method. Make sure to also include the initial condition.
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