An outdoor decorative pond in the shape of a hemispherical tank is to be ­lled with water pumped into the tank through an inlet in its bottom. Suppose that the radius ofthe tank is R = 10 ft, that water is pumped in at a rate of Ï€ft³/min, and that the tank is initially empty. See the following figure. As the tank fi­lls, it loses water 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 is k = 0.01.

(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 equation for the height h of the water at time t. The volume of the water shown in the fi­gure is V = Ï€Rh² €“ 1/3 ph³,

where R = 10. Express the area of the surface of the water A = Ï€r² in terms of h.

(b) Solve the differential equation in part (a). Graph the solution.

(c) If there were no evaporation, how long would ittake the tank to ­fill?

(d) With evaporation, what is the depth of the water at the time found in part (c)? Will the tank ever be fi­lled? Prove your assertion.

Transcribed Image Text:

Output: water evaporates at rate proportional to area A of surface er- Input: water pumped in at rate 7ft/min (a) hemispherical tank (b) cross-section of tank