SOLUTION: A cone is half full in volume, how deep is the water in the cone if it is 12cm in diameter at the top and the total height of the cone is 16cm.

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Question 1205486: A cone is half full in volume, how deep is the water in the cone if it is 12cm in diameter at the top and the total height of the cone is 16cm.
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

OH CRAP!  I misread and thought the container was 12 cm high instead
of 16. So I'll let another tutor do it.  Maybe I'll come back and redo 
it later.




The volume of a cone is V=expr%281%2F3%29pi%2Ar%5E2%2Ah

The radius of the top of the container is 6, since it is
half the diameter, 12.

The volume of the whole cone container = expr%281%2F3%29%2Api%2A6%5E2%2A12

Let the radius of the cone of water = x

Since the height of the cone container is twice the radius, that
will be true for the cone of water, so the height if the cone of
water will be twice x, or 2x.

The volume of the cone of water = expr%281%2F3%29pi%2Ax%5E2%2A2x

Since the volume of the cone of water is half the volume of the cone
container, the former equals twice the latter: 

expr%281%2F3%29pi%2A6%5E2%2A12%22%22=%22%222%2Aexpr%281%2F3%29%2Api%2Ax%5E2%2A2x

cross%28expr%281%2F3%29pi%29%2A6%5E2%2A12%22%22=%22%222%2Across%28expr%281%2F3%29pi%29%2Ax%5E2%2A2x

6%5E2%2A12%22%22=%22%222%2Ax%5E2%2A2x

36%2A12%22%22=%22%224%2Ax%5E3

Divide both sides by 4

9%2A12%22%22=%22%22x%5E3

3%5E2%2A3%2A2%5E2%22%22=%22%22x%5E3

3%5E3%2A2%5E2%22%22=%22%22x%5E3

root%283%2C3%5E3%2A2%5E2%29%22%22=%22%22root%283%2Cx%5E3%29

3%2Aroot%283%2C2%5E2%29%22%22=%22%22x

3%2Aroot%283%2C4%29%22%22=%22%22x

4.762203156%22%22=%22%22x

I'm not sure what you mean by "how deep is the water?"

If it's the height of the cone of water it's = 2x = 9.524406312 cm.

If it's distance from the top of the container to the top
of the water it's 12 - 9.524406312 = 2.475593688 cm.

Round off as you were told.

Edwin

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The container is a cone; and the water in the container forms a cone that is similar to the whole container.

So there is no need in this problem to bother with the formula for the volume of a cone. By a powerful general principle for similar figures, if the scale factor (ratio of linear measurements) between two similar figures is A:B, then the ratio of area measurements between the figures is A^2:B^2, and the ratio of volume measurements between the figures is A^3:B^3.

In this problem, the ratio of volumes of the two cones is 1:2, so the ratio of linear measurements is 1%3Aroot%283%2C2%29

The height of the large cone (the container) is 16cm, so the height of the small cone (the depth of the water) is 16%2Froot%283%2C2%29 = 12.70cm to 2 decimal places.

ANSWER: 12.70cm