smc

In a cone, radius and height each vary inversely with increasing height (i.e. the radius of the cone formed by cutting off the mountain at $4,000$ feet is half that of the original mountain). Therefore, volume varies as the inverse cube of increasing height (expressed as a percentage of the total height of cone): $V_I\times {\text{Height}}^3=V_N$ Plugging in our given condition, $\frac{1}{8}={\text{Height}}^3\Rightarrow \text{Height}=\frac{1}{2}$. $8000\cdot \frac{1}{2}=4000\Rightarrow \boxed{4000}$.