Question

(e) Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height is \( \tan^{-1} (\sqrt{2}) \):

Hint:

Optimization problems often involve substituting constraints into the objective function.

Answer 1

For a cone, volume \( V = \frac{1}{3} \pi r^2 h \). Using the slant height \( l \): \[ r^2 + h^2 = l^2. \] Substitute and differentiate to maximize \( V \). Solve for \( \theta \) where \( \tan \theta = \frac{r}{h} \). The result is: \[ \theta = \tan^{-1} (\sqrt{2}). \]