4. A pond forms as water collects in a conical depression of radius a and depth h. Suppose that water flows in at a constant rate k and is lost through evaporation at a rate proportional to the surface area. (a) Show that the volume V (t) of water in the pond at time t satisfies the differential equation dv/dt = k-an(3a/Th)2/³1/2/3, where a is the coefficient of evaporation. (b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable? (c) Find a condition that must be satisfied if the pond is not to overflow.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
4. A pond forms as water collects in a conical depression of radius a and depth h. Suppose that water flows in at a
constant rate k and is lost through evaporation at a rate proportional to the surface area.
(a) Show that the volume V(t) of water in the pond at time t satisfies the differential equation
dv/dt = k-αn(3a/Th) 2/3√2/3,
where a is the coefficient of evaporation.
(b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable?
(c) Find a condition that must be satisfied if the pond is not to overflow.
Transcribed Image Text:4. A pond forms as water collects in a conical depression of radius a and depth h. Suppose that water flows in at a constant rate k and is lost through evaporation at a rate proportional to the surface area. (a) Show that the volume V(t) of water in the pond at time t satisfies the differential equation dv/dt = k-αn(3a/Th) 2/3√2/3, where a is the coefficient of evaporation. (b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable? (c) Find a condition that must be satisfied if the pond is not to overflow.
Expert Solution
steps

Step by step

Solved in 5 steps

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,