Question

an open box is to be made from a rectangular piece of material 10 inches long and 8 inches wide by cutting equal squares of side $x$ from each corner and turning up the sides. (hint: draw a sketch of the material.)

1. what are the dimensions of the box?
2. write a trinomial expression for the volume of the box.

Explanation:

Step1: Find box length

When we cut a square of side $x$ from both ends of the 10-inch length, the remaining length for the box is $10 - 2x$.

Step2: Find box width

When we cut a square of side $x$ from both ends of the 8-inch width, the remaining width for the box is $8 - 2x$.

Step3: Identify box height

The height of the box is the side length of the cut square, which is $x$.

Step4: Write volume formula

Volume $V$ of a box is $\text{length} \times \text{width} \times \text{height}$. Substitute the dimensions:
$V = x(10-2x)(8-2x)$

Step5: Expand the binomials first

Multiply $(10-2x)(8-2x)$:
$$(10)(8) + 10(-2x) -2x(8) + (-2x)(-2x) = 80 -20x -16x +4x^2 = 4x^2 -36x +80$$

Step6: Multiply by $x$ to get trinomial

$V = x(4x^2 -36x +80) = 4x^3 -36x^2 +80x$

Answer:

1. The dimensions of the box are: Length: $10-2x$ inches, Width: $8-2x$ inches, Height: $x$ inches
2. The trinomial (cubic) expression for volume is $4x^3 -36x^2 +80x$