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Answer :
Sure! Let's go through solving this problem step-by-step:
### Problem Breakdown:
We have a piece of paper that's 8.5 inches by 14 inches, and we want to create an open-top box by cutting squares of side length [tex]\(x\)[/tex] from each corner and folding up the sides. The volume [tex]\(V\)[/tex] of this box can be described as a polynomial function of [tex]\(x\)[/tex]:
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)(x) \][/tex]
### Part 1: Degree and Leading Term of the Polynomial
Step 1: Determine the Degree
To find the degree of the polynomial, we need to expand the expression. Each linear expression [tex]\((14 - 2x)\)[/tex], [tex]\((8.5 - 2x)\)[/tex], and [tex]\(x\)[/tex] has an implicit power of 1. When we multiply these together, the degree of the resulting polynomial is the sum of the powers:
- From [tex]\((14 - 2x)\)[/tex]: degree is 1
- From [tex]\((8.5 - 2x)\)[/tex]: degree is 1
- From [tex]\((x)\)[/tex]: degree is 1
Adding these gives us a degree of 3.
Step 2: Identify the Leading Term
The leading term is the term in the polynomial which has the highest degree. When the expression is fully expanded, the leading term comes from multiplying the highest degree terms from each factor:
- [tex]\(14 - 2x\)[/tex] contributes [tex]\(-2x\)[/tex],
- [tex]\(8.5 - 2x\)[/tex] contributes [tex]\(-2x\)[/tex],
- [tex]\(x\)[/tex] contributes [tex]\(x\)[/tex].
Thus, multiplying these terms: [tex]\((-2x) \cdot (-2x) \cdot x = 4x^3\)[/tex].
So, the leading term of the polynomial is [tex]\(4x^3\)[/tex].
### Part 2: Intercepts of the Graph of [tex]\( V \)[/tex]
Horizontal Intercepts (Roots)
The horizontal intercepts occur where [tex]\( V(x) = 0 \)[/tex]. These correspond to the values of [tex]\(x\)[/tex] that make the expression zero:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(14 - 2x = 0 \Rightarrow x = 7\)[/tex]
3. [tex]\(8.5 - 2x = 0 \Rightarrow x = 4.25\)[/tex]
Since [tex]\(x\)[/tex] represents the side length of the squares cut from each corner, the feasible solutions are those within the constraints of the physical problem, meaning [tex]\(x\)[/tex] must be between 0 and the shortest half-dimension of the paper, which is 4.25 inches. Therefore, the valid horizontal intercepts are [tex]\(x = 0\)[/tex] and [tex]\(x = 4.25\)[/tex].
Vertical Intercept
The vertical intercept is what happens when [tex]\(x = 0\)[/tex]. Substituting [tex]\(x = 0\)[/tex] into the equation gives:
[tex]\[ V(0) = (14 - 2 \times 0)(8.5 - 2 \times 0)(0) = 0 \][/tex]
So, the vertical intercept is also [tex]\(0\)[/tex].
### Do These Points Make Sense?
In the context of this problem, these intercepts make sense:
- At [tex]\(x = 0\)[/tex], no squares are cut out, hence no box is formed (volume is 0).
- At [tex]\(x = 4.25\)[/tex], the maximum feasible square side length is cut out (since cutting larger would result in negative dimensions), and also results in no box being formed (volume is again 0).
In conclusion, the polynomial is of degree 3 with a leading term of [tex]\(4x^3\)[/tex], and the valid intercepts reflect practical constraints.
### Problem Breakdown:
We have a piece of paper that's 8.5 inches by 14 inches, and we want to create an open-top box by cutting squares of side length [tex]\(x\)[/tex] from each corner and folding up the sides. The volume [tex]\(V\)[/tex] of this box can be described as a polynomial function of [tex]\(x\)[/tex]:
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)(x) \][/tex]
### Part 1: Degree and Leading Term of the Polynomial
Step 1: Determine the Degree
To find the degree of the polynomial, we need to expand the expression. Each linear expression [tex]\((14 - 2x)\)[/tex], [tex]\((8.5 - 2x)\)[/tex], and [tex]\(x\)[/tex] has an implicit power of 1. When we multiply these together, the degree of the resulting polynomial is the sum of the powers:
- From [tex]\((14 - 2x)\)[/tex]: degree is 1
- From [tex]\((8.5 - 2x)\)[/tex]: degree is 1
- From [tex]\((x)\)[/tex]: degree is 1
Adding these gives us a degree of 3.
Step 2: Identify the Leading Term
The leading term is the term in the polynomial which has the highest degree. When the expression is fully expanded, the leading term comes from multiplying the highest degree terms from each factor:
- [tex]\(14 - 2x\)[/tex] contributes [tex]\(-2x\)[/tex],
- [tex]\(8.5 - 2x\)[/tex] contributes [tex]\(-2x\)[/tex],
- [tex]\(x\)[/tex] contributes [tex]\(x\)[/tex].
Thus, multiplying these terms: [tex]\((-2x) \cdot (-2x) \cdot x = 4x^3\)[/tex].
So, the leading term of the polynomial is [tex]\(4x^3\)[/tex].
### Part 2: Intercepts of the Graph of [tex]\( V \)[/tex]
Horizontal Intercepts (Roots)
The horizontal intercepts occur where [tex]\( V(x) = 0 \)[/tex]. These correspond to the values of [tex]\(x\)[/tex] that make the expression zero:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(14 - 2x = 0 \Rightarrow x = 7\)[/tex]
3. [tex]\(8.5 - 2x = 0 \Rightarrow x = 4.25\)[/tex]
Since [tex]\(x\)[/tex] represents the side length of the squares cut from each corner, the feasible solutions are those within the constraints of the physical problem, meaning [tex]\(x\)[/tex] must be between 0 and the shortest half-dimension of the paper, which is 4.25 inches. Therefore, the valid horizontal intercepts are [tex]\(x = 0\)[/tex] and [tex]\(x = 4.25\)[/tex].
Vertical Intercept
The vertical intercept is what happens when [tex]\(x = 0\)[/tex]. Substituting [tex]\(x = 0\)[/tex] into the equation gives:
[tex]\[ V(0) = (14 - 2 \times 0)(8.5 - 2 \times 0)(0) = 0 \][/tex]
So, the vertical intercept is also [tex]\(0\)[/tex].
### Do These Points Make Sense?
In the context of this problem, these intercepts make sense:
- At [tex]\(x = 0\)[/tex], no squares are cut out, hence no box is formed (volume is 0).
- At [tex]\(x = 4.25\)[/tex], the maximum feasible square side length is cut out (since cutting larger would result in negative dimensions), and also results in no box being formed (volume is again 0).
In conclusion, the polynomial is of degree 3 with a leading term of [tex]\(4x^3\)[/tex], and the valid intercepts reflect practical constraints.
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