A warehouse having a volume of 10,000 ft^3 and a floor area of 1000 ft^2 is to be built. The cos

step 1 So it is safe to minimize the construction cost. So we need to minimize construction cost. Now t

A warehouse having a volume of 10,000 ft^3 and a floor area...

A warehouse having a volume of $10,000 \mathrm{ft}^3$ and a floor area of $$1000 \mathrm{ft}^2$$ is to be built. The cost of constructing the floor is $$\$ 6.00 / \mathrm{ft}^2$$, the cost of the roof is $$\$ 10.00 / \mathrm{t}^2$$, and the cost of the walls is $$\$ 20.00 / \mathrm{ft}^2$$. If $\mathrm{W}$ is the width, $\mathrm{L}$ the length, and $\mathrm{H}$ the height of the building, what should the dimensions be such that the cost is minimal?

Key Concepts

Cost Function Modeling

Cost function modeling is the process of creating a mathematical representation of the total cost associated with a project. The function typically aggregates the costs of different components or resources, each possibly having its own rate, to determine an expression that can be minimized or maximized in accordance with the project’s requirements.

Constraint Substitution

Constraint substitution is a technique where one or more constraint equations are solved for one variable and then substituted into the objective function. This method reduces the number of independent variables, simplifying the optimization problem and making it more tractable by allowing standard calculus techniques to be applied effectively.

Constrained Optimization

Constrained optimization involves finding the optimum (maximum or minimum) of an objective function while satisfying one or more constraints. It is widely used in problems where design or resource limits, such as fixed volume or area, must be met, which requires the use of strategies to incorporate these constraints directly into the optimization problem.

Lagrange Multipliers

The method of Lagrange multipliers is used to find the local extrema of a function subject to one or more equality constraints. By introducing additional variables (multipliers), the method transforms the constrained optimization problem into a system of equations, where the gradients of the objective function and the constraints are aligned, providing a systematic way to handle complex design constraints.

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