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 dV dt - 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. (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 V(t) = 10(t + 5e-0.1t) - satisfies the ODE and the initial condition. with time: (b) 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 rate out, using either separation o 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 dV dt - 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. (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 V(t) = 10(t + 5e-0.1t) - satisfies the ODE and the initial condition. with time: (b) 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 rate out, using either separation o 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
dV
dt
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.
(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
V(t) = 10(t + 5e-0.1t)
satisfies the ODE and the initial condition.
(b) Now assume that the hole is slowly growing in size, so that the flow rate out increases
with time:
fout (V, t) = 0.1(1 + 0.1t) V.
Solve the ordinary differential equation for V(t) with this flow rate out, using either separation of
variables, or the integrating factor method. Make sure to also include the initial condition.
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