### 6.2: The Return of the Box

Earlier, we learned we can make a box from a piece of paper by cutting squares of side length \(x\) from each corner and then folding up the sides. Let's say we now have a piece of paper that is 8.5 inches by 14 inches. The volume \(V\), in cubic inches, of the box is a function of the side length \(x\) where:

\[ V(x) = (14-2x)(8.5-2x)(x) \]

1. Identify the degree and leading term of the polynomial. Explain or show your reasoning.
2. Without graphing, what can you say about the horizontal and vertical intercepts of the graph of \(V\)? Do these points make sense in this situation?

A sheet of paper measuring 8.5 inches by 14 inches is currently in our possession. The box's volume V, measured in cubic inches, depends on the side length x, where V (x) = (14-2x)(8.5-2x)(x).

So, we have :

V (x) = (14-2x)(8.5-2x)(x)

[tex]V(x) = (14-2x)(8.5-2x)(x).\\V (x) = (14-2x)(8.5x-2x^{2} )\\V(x) = 4x^{3} -45x^{2} + 119x[/tex]

Thus, the leading term of the polynomial is [tex]4x^{3}[/tex] and the degree of the polynomial is 3.

Now, we have :
[tex]V(x) = 4x^{3} -45x^{2} + 119x\\V(x) = x ( 4x^{2} -45x +119)\\V(x) = x (x-7)(4x-17)[/tex]

Now, For intercepts, V(x) = 0

[tex]V(x) = x (x-7)(4x-17) = 0\\x (x-7)(4x-17) = 0\\\\x = 0, 7, 4.25[/tex]

Now when x = 0, V(x) =0

when x = 7, f(x) = 135

when x = 4.25, f(x) = 252.89

So, the points are (7, 135) and ( 4.25, 252.89)

These points represent the value of x and their corresponding value of V.

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JoanBlondell JoanBlondell · Answer 2 · 2024-10-23T20:00:30-04:00

Final answer:

The polynomial is of degree 3 and the leading term is -4x^3. The horizontal intercepts are at x = 0, x = 7, and x = 4.25, representing the lengths of cuts that result in zero volume. The vertical intercept of the graph is at y = 0.

Explanation:

The function V(x) = (14-2x)(8.5-2x)(x) represents the volume of the box. This is a trinomial equation, therefore its degree is 3 and its leading term is -4x^3.

The horizontal intercept (or x-intercept) is the value of x for which V(x) = 0. From the equation, we can see that V(x) = 0 when x = 0, x = 7, and x = 4.25. This implies that there are horizontal intercepts at x = 0, x = 7, and x = 4.25.

As for the vertical intercept (or y-intercept), it is the value of V(x) when x = 0. Substituting x = 0 in the equation, V(x) = (14)(8.5)(0) = 0. So, the vertical intercept is at y = 0. In the context of making a box, these x-intercepts make sense as they represent the possible lengths of cuts for which the box will have a volume of zero.

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