Question 1193372
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The nice thing about this problem is that you don't need to deal with the formula for the volume of a frustum, or with ANY geometric formula.<br>
If the volume of the frustum is 3/5 of the volume of the whole cone, then the small cone that is cut off by the plane is 2/5 of the volume of the whole cone.<br>
The two cones are similar, so the ratio of the volumes is the cube of the ratio of the heights.  Knowing that the volume of the small cone is 2/5 of the volume of the whole cone, the ratio of the heights of the two cones is<br>
{{{(2/5)^(1/3)}}} = 0.73681 to 5 decimal places.<br>
Then, since the height of the small cone is 0.73681 times the height of the whole cone, the height of the frustum is 1-0.73681 = 0.26319 times the height of the whole cone.<br>
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Clarification for the student who asked a question about my response....<br>
The ratio of the volumes is the cube (3rd power) of the ratio of the heights; so the ratio of the heights is the CUBE ROOT (1/3 power) of the ratio of the volumes.<br>
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NO!!!<br>
The problem has nothing to do with the formula for the volume of a cone (which happens to contain a factor of 1/3).<br>
The 1/3 in the problem is an exponent.  The ratio of heights of the two cones is the cube root (1/3 power) of the ratio of the volumes.<br>