Question

you make a tin box by cutting x - inch - by - x - inch pieces of tin off of the corners of a rectangle and folding up each side. the plan for your box is shown.
a what are the dimensions of the original piece of tin?
(12 - 2x) by (6 - 2x)
b write a function that represents the volume v of the box. without multiplying, determine its degree.
(12 - 2x)(6 - 2x)x

Explanation:

Step1: Recall volume formula for a rectangular - box

The volume $V$ of a rectangular - box is given by $V = l\times w\times h$, where $l$ is the length, $w$ is the width, and $h$ is the height.

Step2: Identify the dimensions of the box

After cutting $x$-inch squares from the corners and folding up the sides, the length $l = 12 - 2x$, the width $w=6 - 2x$, and the height $h = x$.

Step3: Write the volume function

The volume function $V(x)=(12 - 2x)(6 - 2x)x$.

Step4: Determine the degree of the polynomial

The degree of a polynomial is the highest power of the variable. Expand the product: $(12 - 2x)(6 - 2x)x=(72-24x - 12x + 4x^{2})x=(72-36x + 4x^{2})x=72x-36x^{2}+4x^{3}$. The highest - power of $x$ is 3, so the degree of the polynomial is 3.

Answer:

The volume function is $V(x)=(12 - 2x)(6 - 2x)x$ and its degree is 3. The dimensions of the original piece of tin are 12 inches by 6 inches.