64. (The clepsydra, or water clock) A 12 hour water clock is to be designed with the dimensions shaped like the surface obtained by revolving the curve y = f(x) around the y-axis. What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at the constant rate of 4 inches per hour?
Height = 4 ft
Radius of the top of the clock = 1 ft
Hint: A(y) dy/dt = - k* sqrt(y)
43. Arthur Clarke's The Wind from the Sun describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.0001 g = 0.0098 m/s2. Suppose this spacecraft starts from rest at time t = 0 and simultaneously fires a projectile (straight ahead in the same direction) that travels at one-tenth of the speed c = 3 x 108 m/s of light. How long will it take the spacecraft to catch up with the projectile, and how far will it have traveled by then?
The solution shows how to set up and solve the differential equation in two practical scenarios.