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Let h(t) and V(t) be the height and volume of water in a tank at time t

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Let h(t) and V(t) be the height and volume of water in a tank at time t. If water drains through a hole with area α at the bottom of the tank, then Torricelli's Law says that

                                                dV/dt= -α√2gh

Where g is the acceleration due to gravity. So the rate at which water flows from the tank is proportional to the square root of the water height.

Part 1

Suppose the tank is cylindrical with height 6 feet and radius 2 feet and the hole is circular with radius 1 inch. If we take g=32ft/s2, show that h satisfies the differential equation

 

                                                       dh/dt= -1/72√h

You are going to want to start with the formula for volume of a cylinder then use the equation given above for Torricelli's Law to substitute in. Your work should start with the volume formula and end with the differential equation above.

Part 2

Because of the rotation and the viscosity of the liquid, the theoretical model given by our first equation isn't quite right. Instead, the model

                                          dh/dt=k√h

Is often used and the constant k is determined experimentally. You are going to use the video in this lesson to determine the value for k for water draining from a 4 mm hole into a 2 liter bottle, an expression for h(t), and the amount of time it takes the water level to go from 10 cm to 0 cm.

a) To determine the value of k, you are going to want to use differential equations on the model above. You are encouraged to use the initial condition to solve for your constant. You are going to want to use at least one point for a second coordinate from the video. You will get this point from pausing the video intermittently.

b) To find an expression for h(t), substitute your value for k into the differential equation you found.

c) To determine the amount of time it takes for the water to completely empty, use your equation for h(t) and solve for t.