SOLUTION: when conical bottle rests on its base, the water in the bottles is 8cm from the vertex. When the same conical bottle is turned upside down, the water level is 2cm from its base. Wh

Question 1133099: when conical bottle rests on its base, the water in the bottles is 8cm from the vertex. When the same conical bottle is turned upside down, the water level is 2cm from its base. What is the height of the bottle?

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Let the radius and height of the cone be r and x. The volume of the cone is then

When the cone is resting on its base, the water is in the shape of a frustum of a cone, with the surface of the water 8cm from the vertex.

View the volume of water as the volume of the whole cone, minus the volume of the cone that is NOT filled with water.

Those two cones are similar; their heights are in the ratio 8:x; so their volumes are in the ratio (8:x)^3 = 512:x^3. Then the fraction of the volume of the whole cone that is filled with water is

When the cone is resting on its vertex, the water is in the shape of a cone that is similar to the whole cone, with the surface of the water 2cm from the base.

The ratio of the heights of those two cones is (x-2):x; so their volumes are in the ratio (x-2)^3:x^3. The fraction of the volume of the whole cone that is filled with water now is

We have two expressions that are both the fraction of the whole cone that is filled with water; so those expressions must be equal.

The quadratic does not factor; the quadratic formula gives x = 1+sqrt(85) = 10.220 to 3 decimal places.

ANSWER: The height of the cone, to 3 decimal places, is 10.220cm.
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