Question

Evaporation Problem

<p>An outdoor decorative pond in the shape of a hemispherical tank is to be filled with water pumped into the tank through an inlet in its bottom. Suppose thatthe radius of the tank is R=10 ft, that water is pumped in at a rate of (pi) feet cubed per minute, and that the tank is initially empty. AS the tank fills, it loseswater through evaporation. Assume that the rate of evaporation is proportional to the area A of the surface of the water and that the constant of proportionality isk=0.01.<br /><br />a) the rate of change dV/dt of the volume of the water at time t is a net rate. Use this net rate to determine a differential equationfor the height h of the water at time t. The volume of the water shown in the figure is V=pi*R*h^2-(1/3)*pi*h^3, where R=10 ft. Express the area of the surface of thewater A=pi*r^2 in terms of h.<br /><br />b) Solve the differential equation in part (a). Graph the solution.<br /><br />c) If there were noevaporation, how long would it take the tank to fill?<br /><br />d) With evaporation, what is the depth of the water at the time found in part (c)? Willthe tank ever be filled? Prove your assertion.</p>

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