A box (with no top) is to be constructed from a piece of...

A box (with no top) is to be constructed from a piece of cardboard with sides of length $A$ and $B$ by cutting out squares of length h from the corners and folding up the sides (Figure 30). Find the value of $h$ that maximizes the volume of the box if $A=15$ and $B=24$. What are the dimensions of this box?

A box (with no top) is to be constructed from a piece of cardboard with sides of length $A$ and $B$ by cutting out squares of length h from the corners and folding up the sides (Figure 30).
Find the value of $h$ that maximizes the volume of the box if $A=15$ and $B=24$. What are the dimensions of this box?

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Key Concepts

Constraint Analysis

This involves understanding the limitations imposed on the variables based on the physical context of the problem. In the case of constructing a box, this means ensuring that the dimensions remain positive and valid after modifications, which restricts the possible values of the variable representing the size of the cut-out squares.

Differentiation

Differentiation is a key tool in calculus used to compute the derivative of a function, which indicates the rate of change of the function with respect to its variable. In optimization problems, setting the derivative equal to zero helps identify the critical points where the function may have a maximum or minimum value.

Optimization in Calculus

This concept involves finding the maximum or minimum values of a function by analyzing its derivatives. In problems like these, calculus is used to determine the point at which a given function, such as a volume function, attains its maximum or minimum value, subject to certain conditions or constraints.

Volume Function Modeling

This is the process of expressing the volume of the constructed object in terms of one or more variables. By relating the dimensions of the box (which change when squares are cut out and sides are folded) to a variable, one can derive an equation for the volume that can be optimized.

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