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A cone-shaped container is oriented with its circular base on the top and its apex at the bottom. It has a radius of 18 inches and a height of 6 inches. The cone starts filling up with water. What fraction of the volume of the cone is filled when the water reaches a height of 2 inches?

 Apr 7, 2020
edited by GAMEMASTERX40  Apr 7, 2020
 #1
avatar+932 
+2

The water will reach a height of 2 inches creating a similar cone. Since the ratio of the height of the smaller to the larger cone is 2:6 or 1:3, that means that the radius of the cones are in a ratio of 1:3 meaing the smaller cone has a radius of 6. Knowing this, the volume of a cone is 1/3Bh, where B is area of base and h is the height so we get 1/3*36pi*2= 24 pi.

 

Hope it helps!

 Apr 7, 2020
 #2
avatar+932 
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The fast way to do it is to realize that the ratio of corresponding side lengths is 1 to 3. The ratio of volumes of similar shapes is ration of sides cubed. So the ratio of the volumes would be 1^3 to 3^3 or 1 to 27.

So if the large cone has a volume of 648pi, then 648pi/27= 24pi. This is a neat trick to know and works with area as well.

 

Hope it helps!

 Apr 7, 2020
 #3
avatar+111329 
+2

Note  that  the  small  cone formed  by the water  is similar to the  larger cone

 

The scale  factor  for the small cone to the large cone =   2 / 6  =  1/3

 

The  fraction of the  volume of the cone when  filled to 2 in = scale factor^3 =  (1/3)^3  =  1/27

 

cool cool cool   

 Apr 7, 2020

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