Name: optimizing silo design with lagrange multipliers
Uploaded: 2026-06-08T09:38:16.812Z
Duration: 1 min 30 s
Channel: JoVE

Designing an efficient storage silo requires maximizing capacity while limiting material use. Consider a silo with a circular cylinder, a flat base, and a hemispherical roof. The goal is to maximize the total volume while keeping the surface area fixed, which helps reduce material cost.Let the base radius be r, and the cylinder height be h. The total volume equals the cylinder&#x2019;s volume plus the hemisphere&#x2019;s volume.The total surface area includes the flat base, the cylindrical side, and the hemispherical roof. Since the surface area is fixed, the volume is constrained.To handle this restriction, the Lagrange multiplier method introduces a variable, lambda. This method links the gradient of the volume to lambda times the gradient of the surface area.First, differentiating with respect to h gives an expression for lambda in terms of r. Then, differentiating with respect to r and substituting lambda gives a relationship between h and r.The result shows that the maximum volume happens when the cylinder height equals its radius.This condition balances the cylindrical and hemispherical parts and maximizes the volume for the fixed surface area.

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&#x2014;i.e., a fixed surface area&#x2014;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 &#x3BB; in terms of the radius. A second derivative condition, taken with respect to the radius, introduces both geometric variables. Eliminating &#x3BB; 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.

Lagrange Multipliers: Problem Solving

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JoVE Core

Calculus

Unit

Partial Derivatives and Gradients

KEY TERMS AND CONCEPTS

optimizing silo design with lagrange multipliers

A silo with a cylindrical base and a hemispherical roof can be sized with Lagrange multipliers. This calculus method helps find the best design when the surface area must stay fixed. In this problem, the goal is to maximize storage volume while using a set amount of material.

The model uses two main variables: the radius r of the base and the height h of the cylindrical section. The total volume comes from adding the volume of the cylinder and the hemisphere. The fixed surface area includes the flat circular base, the curved side of the cylinder, and the curved surface of the hemisphere. Because the surface area is constrained, the dimensions cannot be chosen freely.

To handle that restriction, the Lagrange multiplier method is applied. The condition for an extremum is written as ∇V = λ∇s, where V is volume and s is surface area. This means that, at the best design, the rate of change of volume is proportional to the rate of change of surface area. Differentiating with respect to the cylinder height gives an expression for λ in terms of the radius.

A second derivative step with respect to the radius brings both geometric variables into the same relation. Eliminating λ from the two equations gives a simple proportion for the silo. The result is h = r. So, the most efficient silo has a cylindrical height equal to the radius of its base. This balance between the cylinder and the hemispherical roof gives the maximum enclosed volume for the available construction material.

Optimizing Silo Design with Lagrange Multipliers

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A silo with a cylindrical base and a hemispherical roof can be sized with Lagrange multipliers. This calculus method helps find the best design when the surface area must stay fixed. In this problem, the goal is to maximize storage volume while using a set amount of material. The model uses two main variables: the radius r of the base and the height h of the cylindrical section. The total volume comes from adding the volume of the cylinder and the hemisphere. The fixed surface area includes...

key terms and concepts

Use Lagrange multipliers to optimize a silo’s dimensions with fixed surface area. Learn why the best design has cylinder height equal to the base radius.

Optimizing Silo Design with Lagrange Multipliers | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Optimizing Silo Design with Lagrange Multipliers

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