Consider the initially full hemispherical water tank of Example 8, except that the radius r of its circular bottom hole is now unknown. At 1 P.M. the bottom hole is opened and at 1:30 P.M. the depth of water in the tank is 2 ft.

(a) Use Torricelli’s law in the form dV/dt = (-0.6) π r² √2gy (taking constriction into account) to determine when the tank will be empty.

(b) What is the radius of the bottom hole?

Data from in Example 8

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The differential equation y³ dx + 3xy2 dy = 0 (25) is exact because we can immediately see that the function F(x, y) = xy3 has the property that Fx = y³ and Fy = 3xy2. Thus a general solution of Eq. (25) is xy³ = C; if you prefer, y(x) = kx-1/3