Lagrange Multipliers: Problem Solving

A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.

The total volume of the silo is obtained by adding the volume of the cylindrical section to that of the hemispherical roof, while the fixed surface area includes the flat circular base, the curved lateral surface of the cylinder, and the curved surface of the hemisphere. Because the surface area is fixed, the dimensions are constrained. To account for this restriction, the Lagrange multiplier method is applied. In this approach, the condition for an extremum is written as

\begin{equation*}\nabla V = \lambda \nabla s\end{equation*}

This relation states that, at the optimal design, the rate of change of volume is proportional to the rate of change of surface area.

Differentiating with respect to the cylindrical height first produces an expression for the multiplier λ in terms of the radius. A second derivative condition, taken with respect to the radius, introduces both geometric variables. Eliminating λ between these two relations yields a simple proportion between the dimensions of the silo.

The resulting condition is

\begin{equation*}h=r\end{equation*}

Thus, the most efficient silo is obtained when the height of the cylindrical section equals the radius of the base. This proportion balances the contributions of the cylindrical and hemispherical parts, yielding the maximum enclosed volume for a given amount of construction material.