Question 1198244
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The altitude of a pyramid is 12 cm. A parallel to the base cut the pyramid into two solids of equal volumes. 
The distance of the cutting plane from the vertex is expressed as d = α^3√β cm where α and β are integers. 
Find the smallest sum of α and β.
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                Your formulation in the post is INCORRECT.
                I was so kind that re-wrote the problem in a right form.
                The correct version is as follows:



<pre>
    The altitude of a pyramid is 12 cm. A parallel to the base cut the pyramid into two solids of equal volumes. 
    The distance of the cutting plane from the vertex is expressed as d = {{{alpha*root(3,beta)}}} cm, where α and β are integers. 
    Find the smallest sum of α and β.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;See my solution below for this corrected version.



<pre>
Let x be the distance from the vertex to the cutting plane.


The cut part and the whole pyramid are similar solids, so the ratio
of their volumes (which is 1:2) is the cube of the ratio of their 
respective linear elements.


It gives us this proportion

    {{{x^3/12^3}}} = {{{1/2}}},

or

    x^3 = {{{12^3/2}}} = {{{(3^3*4^3)/2}}} = {{{(3^3*2^6)/2}}} = {{{3^3*2^3*4}}}.


Hence,  x = {{{3*2*root(3,4)}}} = {{{6*root(3,4)}}}.


Thus {{{alpha}}} = 6,  {{{beta}}} = 4, and the sum  {{{alpha}}} + {{{beta}}} = 6 + 4 = 10.    <U>ANSWER</U>
</pre>

Solved.