Solve each problem. A rectangular piece of metal is 10 in....

Solve each problem. A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. If the volume of the box is $832 \mathrm{in} .^{3}$, what were the original dimensions of the piece of metal?

Solve each problem.
A rectangular piece of metal is 10 in. longer than it is wide. Squares with sides 2 in. long are cut from the four corners, and the flaps are folded upward to form an open box. If the volume of the box is $832 \mathrm{in} .^{3}$, what were the original dimensions of the piece of metal?

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Key Concepts

Geometric Modeling

This concept involves translating a real-world situation into a geometric framework by recognizing how the original shape is altered through operations like cutting and folding. It includes understanding how the dimensions of the original shape change when parts are removed and transformed into a three-dimensional object.

Variable Definition and Algebraic Representation

Identifying and defining variables for the unknown quantities allows the problem to be expressed in mathematical terms. This process involves setting up equations based on the relationships given in the problem, such as the fixed difference in dimensions and the changes after cutting squares from the corners.

Volume Calculation

This involves using the formula for the volume of a box, which is the product of its length, width, and height. In problems like this, one must correctly express these dimensions in terms of the original measurements after accounting for the modifications made to the shape.

Quadratic Equations

Algebraic manipulation often leads to quadratic equations when relationships involve products and sums of dimensions. Solving these quadratic equations is essential in finding the correct numerical values for the variables that describe the dimensions of the initial shape.

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