If the height of a cone is decreased by 20% whereas its radius is increased by 25% , can we find the percentage increase in total surface of the cone ?

With the above information , I am trying to find the answer in the traditional way using formulae, but I am not able to arrive at the answer. Is the data inconsistent in the above question ? If yes, can you explain why ?

  • $\begingroup$ Huh. Doesn't look like the percentage is going to be a constant. $\endgroup$ – Mike Jul 4 '12 at 10:25

The surface area of a (right, circular) cone with radius $r$ and slant height $\ell$ is given $A = \pi r^2 + \pi r \ell~$. Given the height $h$ instead, then we know $r^2 + h^2 = \ell^2~$.

The issue here is that neither expression is linear in the desired variables. So, there's no guarantee that if we have a new cone with radius $r'$, height $h'$ proportional to $r,h$ respectively, that $\ell'$ will be proportional to $\ell$ or that $A'$ will be proportional to $A$. The slant height and area will change proportionally if and only if the radius and height stretch by the same factor (or, to use language consistent with the question, increase by the same percentage).


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