Mathematics

A cone's volume depends on both its height and base radius, but they're linked by the fixed slant height constraint $\ell = 3$. The key insight is to express volume in terms of just one variable using the Pythagorean relationship $\ell^2 = h^2 + r^2$, then optimize by finding where $\frac{dV}{dh} = 0$. This transforms a constrained geometry problem into a single-variable calculus problem.

A cone's volume depends on both its height and base radius, but they're linked by the fixed slant height constraint $\ell = 3$. The key insight is to express volume in terms of just one variable using the Pythagorean relationship $\ell^2 = h^2 + r^2$, then optimize by finding where $\frac{dV}{dh} = 0$. This transforms a constrained geometry problem into a single-variable calculus problem.