An open box is to be made from a square piece of cardboard...

An open box is to be made from a square piece of cardboard having dimensions 30 inches by 30 inches by cutting out squares of area $x^{2}$ from each corner, as shown in the figure. (FIGURE CAN'T COPY) a. Express the volume $V$ of the box as a function of $x$ b. State the domain of $V$

An open box is to be made from a square piece of cardboard having dimensions 30 inches by 30 inches by cutting out squares of area $x^{2}$ from each corner, as shown in the figure. (FIGURE CAN'T COPY)
a. Express the volume $V$ of the box as a function of $x$
b. State the domain of $V$

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Key Concepts

Domain of a Function

In physical problems, the domain represents all the possible values of x that produce a valid (i.e., physically meaningful) box. This means ensuring that all dimensions (length, width, and height) are positive. The constraints derived from the dimensions of the cardboard result in a domain for x that prevents any negative or zero dimensions.

Function Modeling

This concept involves translating a real-world scenario into a mathematical expression. In the context of the box problem, it means determining how the dimensions of the box are affected by the variable x (the side-length of the squares removed) and expressing the volume as a function of x.

Volume Calculation

The volume of a three-dimensional object like an open box is calculated as the product of its length, width, and height. For the box formed after cutting squares from a larger piece of cardboard, the new length and width are reduced by twice the size of x, and the height becomes x. This leads to the volume function V(x) = x(30-2x)².

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