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
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.
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