Question

For a cone, if the radius was quadrupled and the slant height was reduced to 1/6 of its original size what would be the formula to find the modified surface area.

Answer 1

Let $r$ be the radius of the cone's base and $\ell$ the slant height.

The surface area of a cone is

$A=\pi r^2+\pi r\ell$

If the radius is quadrupled ($r\to4r$) and the slant height scaled by a factor of $\dfrac16$, then the new area is

$A=\pi(4r)^2+\pi (4r)\left(\dfrac\ell6\right)$
$A=16\pi r^2+\dfrac23\pi r\ell$