A box, open at the top, is to be made from a square piece of...

A box, open at the top, is to be made from a square piece of cardboard by cutting a square out of each corner and turning up the sides. In FIGURE $4.8 .16,$ the white squares are cut out and the cardboard is folded along the dashed lines. Given that the cardboard measures $40 \mathrm{~cm}$ on a side, find the dimensions of the box that will give the maximum volume. What is the maximum volume?

Key Concepts

Critical Point Analysis

After identifying potential optimal values by setting the derivative equal to zero, critical point analysis involves evaluating these points—often using second derivative tests or comparing function values—to determine whether they correspond to a maximum or minimum in the context of the problem.

Domain Restrictions

In practical optimization problems, it is essential to consider the physical constraints of the problem. For instance, the variable representing the size of the cut squares must be within a specific range to ensure that the dimensions of the resulting box are meaningful. These constraints are considered when determining the valid solutions for the maximum volume.

Volume Modeling

This concept involves expressing the volume of the resulting box as a function of a variable—in this case, the side length of the squares removed from each corner. The function is constructed by considering the altered dimensions after the squares are removed and the sides are folded up, leading to a formula that relates the cutting size to the volume of the box.

Optimization using Derivatives

When a problem requires finding the maximum or minimum value of a function, calculus techniques—specifically differentiation—are used. By taking the derivative of the volume function with respect to the variable and setting it equal to zero, one identifies the critical points. These points are candidates for maximum or minimum values, which can be confirmed by further testing or analysis.

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