Question

A candle in the shape of a circular cone has a base of radius r and a height of h that is the same length as the radius. Which expresses the ratio of the volume of the candle to its surface area (including the base)?

Answer 1

\[\frac{ r(1 - \sqrt{2}) }{ -3 }\]

Answer 2

curved length l=sqrt(r^2+h^2)=sqrt(r^2+r^2)=rsqrt2
surface area s=pi r l+pi r^2=pi r(l+r)=pi r(r sqrt 2+r)=pi r^2(sqrt2+1)
volume v =1/3 pi r^2 h=1/3 pi r^3 (h=r)
v/s=(1/3 pi r^3)/{pi r^2(sqrt 2+1)=r/{3(sqrt2+1) *(sqrt2-1)/(sqrt2-1)
=r(sqrt2-1)/3(2-1)
=r(sqrt2-1)/3