ratio of surface areas ,

Geometry Level 3

The height of the right circular cylinder shown above is twice the length of its base diameter. If the largest right circular cone is cut from this cylinder, what is the ratio of the total surface area of the cone to the total surface area of the cylinder. Give your answer as a decimal number rounded to three decimal places.


The answer is 0.512.

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1 solution

Tom Engelsman
Oct 13, 2018

If h h be the height and 2 r 2r be the diameter of the cylinder, then we have the following total surface area formulas:

A c y l i n d e r = 2 π r h + 2 π r 2 = 2 π r ( 2 ⋅ 2 r ) + 2 π r 2 = 10 π r 2 A_{cylinder} = 2\pi rh + 2\pi r^2 = 2\pi r (2 \cdot 2r) + 2\pi r^2 = 10\pi r^2

A c o n e = π r r 2 + h 2 + π r 2 = π r r 2 + ( 2 ⋅ 2 r ) 2 + π r 2 = ( 17 + 1 ) π r 2 A_{cone} = \pi r \sqrt{r^2 + h^2} + \pi r^2 = \pi r \sqrt{r^2 + (2 \cdot 2r)^2} + \pi r^2 = (\sqrt{17} + 1) \pi r^2

Thus the desired ratio computes to:

A c o n e A c y l i n d e r = 17 + 1 10 . \large \frac{A_{cone}}{A_{cylinder}} = \boxed{ \frac{\sqrt{17} + 1}{10}}.

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