Water in a conical tank

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Question from Greg, a student:

Joe is conducting an experiment to study the rate of flow of water from a conical tank.
The dimensions of the conical tank are:
Radius at the initial water level = 13.7 cm
Radius at the reference point = 12.8 cm
Initially the tank is full of water. There is a circular orifice at the bottom of the conical tank with a diameter of 0.635 cm. The water drains from the conical tank into an empty cylindrical tank lying on its side with a radius of 0.500 ft and a length L (ft).

Joe observed the water discharged with an average velocity of 1.50 m/s as the water level lowered from the initial height of 14.0 cm to 5.00 cm in the conical tank. Answer the following:
1. If the initial height of water in the conical tank is 14.0 cm (measured from the reference point, see Fig. 1), how long in seconds will it take for the water level to drain to a height of 5.00 cm?? NOTE: Height refers to the vertical height.

What formula would I use to find out how long in seconds it takes for the water level to drop?

Hi Greg,

The stream of water flowing out of the tank is in the shape of a circular cylinder of radius 0.635/2 cm. The volume of a circular cylinder is given by V = π r² h where in your case r = 0.635/2 cm and V and h are functions of time. Thus, differentiating both sides

V'(t) = π r² h'(t)

and you know that h'(t) = 1.50 m/s so this allows you to calculate V'(t).

Now notice that the rate at which the volume of water flowing out of the tank is the rate at which the volume of water on the conical tank is decreasing. Hence the amount of water in the conical tank is changing at the rate of -V'(t) m/s. The volume of a cone is 1/3 π r² h where r is the radius and h is the height. Again in your case r and h are functions of time.

You didn't send us Fig. 1 but I expect there is enough information in the figure to find a relationship between r and h, probably using similar triangles. This will allow you to write the volume of water in the tank as a function of one variable, either r or h. which should then allow you to complete the problem.

Harley

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