Question

The clepsydra, or water clock, was a device that the ancient Egyptians, Greeks, Romans, and Chinese used to measure the passage of time by observing the change in the height of water that was permitted to flow out of a small hole in the bottom of a container or tank.

Answer 1

Suppose a tank is made of glass and has the shape of a right-circular cylinder of radius 1 ft. Assume that h(0) = 2 ft corresponds to water filled to the top of the tank, a hole in the bottom is circular with radius in., g = 32 ft/s2, and c = 0.6. Use the differential equation in Problem 12 to find the height h(t) of the water.

Answer:

Height of the water = √(128)/147456 ft

Explanation:

Given

Radius, r = 1 ft

Height, h = 2 ft

Radius of hole = 1/32in

Acceleration of gravity, g = 32ft/s²

c = 0.6

Area of the hold = πr²

A = π(1/32)² ---- Convert to feet

A = π(1/32 * 1/12)²

A = π/147456 ft²

Area of water = πr²

A = π 1²

A = π

The differential equation is;

dh/dt = -A1/A2 √2gh where A1 = Area of the hole and A2 = Area of water

A1 = π/147456, A2 = π

dh/dt = (π/147456)/π √(2*32*2)

dh/dt = 1/147456 * √128

dh/dt = √128/147456 ft

Height of the water = √(128)/147456 ft