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If a smaller cone is inscribed in a larger cone as shown, then what will be the radius of the smaller cone if it has the maximum volume?

Attempt
I know that the volume of a cone =$\dfrac{1}{3}\pi r^2h$ , and the maximum volume can found by setting the derivative equal to zero to see where the maximum lies.

I tried to find a relation between the the height of the small cone and the larger cone to express h in terms of r in the equation of volume, but I got nothing.

Help.

Someone
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Maher
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1 Answers1

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You are on the right track. Let $h$ be the height of the small cone. Slice the cones with a vertical plane through the axis. Each cone becomes an isosceles triangle. If you draw the axis you have two right triangles. Use the known shape of the large triangle to get the base of the small triangle, which is the diameter of the small cone, as a function of $h$

Ross Millikan
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  • here is the point.. How to get the base of the small triangle ? – Maher Dec 29 '14 at 19:27
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    Given $h$, you can determine the distance down from the tip of the big cone. The radius of the big cone is linearly proportional to the distance down from the top. The radius of the small cone at its top is the same as the radius of the big cone at the same altitude. Have you drawn the picture that is suggested by my answer? – Ross Millikan Dec 29 '14 at 21:27
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    Yes Yes ! Got it. The answer must be $10/3$. Thanks. – Maher Dec 29 '14 at 22:00