Quandaries and Queries


A vertically inverted cone( i.e. vertex down) has a radius 7
inches and height 24 inches. Water is filled to one third of its
height .Find the ht of water when cone is turned upside down



When I read this problem I saw a conical shaped paper cup, one-third full of water. When you turn it over all the water runs out so the height s zero. But this is probably not the answer you want.

The important expression is that the volume of a cone is given by

volume = 1/3 pi r2 h

where r is the radius of the base and h is the height. You also have to use some information about similar triangles. Below is a sketch of one cone inside another as you have in your problem.

The triangle ABC and EDC are similar so


Using the volume expression, the volume of your cone is

cone volume = 1/3 pi 7 2 24 = 392 pi cubic inches.

Use the similar triangle expression to find a and calculate the volume of the water in the cone by

water volume = 1/3 pi a 2 b = 1/3 pi a 2 8 cubic inches.

The volume of air in the cone is (cone volume) - (water volume).

Now turn the diagram upsidedown. Again you have similar triangles.

This time


Insert the values of h and r and solve for c in terms of d. Use the expression for the volume of a cone to write the volume of the air in terms of c and d. Use the volume for the air you found earlier and the expression for c in terms of d to form an equation in one variable d. Solve for d.



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