To make boxes, a manufacturer cuts equal-sized squares from...

To make boxes, a manufacturer cuts equal-sized squares from each corner of the 10 in. $\times 12$ in. piece of cardboard shown below and then folds up the sides. The polynomial function $f(x)=4 x^{3}-44 x^{2}+120 x$ gives the volume (in cubic inches) of the resulting box when a square with sides $x$ inches long is cut from each corner. Find the volume of a box if 3 -inch squares are cut out. (PICUTURE NOT COPY)

Key Concepts

Algebraic Modeling

Algebraic modeling uses algebraic expressions and equations to represent real-world scenarios. In the context of constructing a box by cutting and folding a piece of material, algebraic modeling translates the physical modifications into a mathematical function, making it easier to analyze and optimize properties like volume.

Volume Calculation in Geometric Problems

Volume calculation is the process of determining the space occupied by a three-dimensional object. In geometric problems involving the formation of objects like boxes, modifying dimensions due to alterations (like cutting squares from corners) is modeled with algebraic formulas to compute the resulting volume accurately.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole-number exponents and multiplied by coefficients, summed together. They are fundamental in algebra and are often used for modeling various real-world situations, making it possible to represent complex relationships in a structured and analyzable way.

Substitution in Functions

Substitution involves replacing the variable in a function with a specific value to evaluate the function. This technique is crucial when determining the result of a polynomial function for particular instances, allowing for direct computation of outcomes such as dimensions or volume in geometric models.

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