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A cylinder of radius 6cm and height 6cm fits perfectly inside a cone, leaving a constant ring of width x around the base of the cylinder. Find the height of the cone. Here is an image, showing the problem:enter image description here

I can guess that pythagorus may have to be used, but I am not sure if I need to know some property about ratios of the sides or something!

Many thanks.

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    $\begingroup$ Similar triangles, $6+x$ and $h$ for the big one, $6$ and $h-6$ for the small one.... $\endgroup$ – Ned Nov 23 '19 at 19:00
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There is no need to use Pythagoras. If you would draw a cross section of the cone, you would see triangles that are similar (not congruent, because of different size). Therefore, we know that the ratio of the sides of these triangles are equal:

$$ \frac{6}{x} = \frac{h}{x+6}. $$

From this, it simply follows that:

$$ h=6+\frac{36}{x}. $$

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  • $\begingroup$ Oh ok. I always fail to spot similar triangles! I understand now though, thanks. $\endgroup$ – Jamminermit Nov 24 '19 at 9:19

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