Area of covering material:

$A = \pi rL \,\, $ where $L = \sqrt{h^2 + r^2}$

$A = \pi r\sqrt{h^2 + r^2} \,\, $

$\dfrac{dA}{dr} = \pi r \left( \dfrac{2h\dfrac{dh}{dr} + 2r}{2\sqrt{h^2 + r^2}} \right) + \pi \sqrt{h^2 + r^2} = 0$

$hr \dfrac{dh}{dr} + r^2 + (h^2 + r^2) = 0$

$\dfrac{dh}{dr} = - \dfrac{2r^2 + h^2}{rh}$

Volume of tepee:

$V = \frac{1}{3} \pi r^2 h$

$\dfrac{dV}{dr} = \frac{1}{3} \pi \left( r^2 \dfrac{dh}{dr} + 2rh \right) = 0$

$r \dfrac{dh}{dr} + 2h = 0$

$r\left( - \dfrac{2r^2 + h^2}{rh} \right) + 2h = 0$

$-2r^2 - h^2 + 2h^2 = 0$

$h^2 = 2r^2$

$h = \sqrt{2}\,r$

$\text{height } = \sqrt{2} \times \text{ radius}$ *answer*