Answer :
Sure, let's break down the problem step by step.
### Given Information
We have a piece of paper that is 8.5 inches by 14 inches. We cut squares of side length [tex]\( x \)[/tex] from each corner and fold up the sides to form a box. The volume [tex]\( V(x) \)[/tex] of the box is given by:
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)x \][/tex]
We need to:
1. Identify the degree and leading term of the polynomial [tex]\( V(x) \)[/tex].
2. Discuss the horizontal and vertical intercepts of the graph of [tex]\( V(x) \)[/tex] and verify if they make sense in this context.
### Question 1: Degree and Leading Term
To identify the degree and leading term:
1. Expand the polynomial [tex]\( V(x) \)[/tex]
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)x \][/tex]
First, multiply the two binomials:
[tex]\[ (14 - 2x)(8.5 - 2x) \][/tex]
[tex]\[ = 14 \cdot 8.5 - 14 \cdot 2x - 2x \cdot 8.5 + 2x \cdot 2x \][/tex]
[tex]\[ = 119 - 28x - 17x + 4x^2 \][/tex]
[tex]\[ = 119 - 45x + 4x^2 \][/tex]
Now, multiply this result by [tex]\( x \)[/tex]:
[tex]\[ V(x) = (119 - 45x + 4x^2)x \][/tex]
[tex]\[ = 119x - 45x^2 + 4x^3 \][/tex]
2. Identify the degree of the polynomial
The degree is the highest power of [tex]\( x \)[/tex]. Here, the highest power is [tex]\( x^3 \)[/tex], so the degree of the polynomial is 3.
3. Determine the leading term
The leading term is the term with the highest power of [tex]\( x \)[/tex]. In this case, it is [tex]\( 4x^3 \)[/tex].
### Answer to Question 1:
- The degree of the polynomial [tex]\( V(x) \)[/tex] is 3.
- The leading term is [tex]\( 4x^3 \)[/tex].
### Question 2: Horizontal and Vertical Intercepts
1. Horizontal intercepts (where [tex]\( V(x) = 0 \)[/tex])
To find the horizontal intercepts, solve [tex]\( V(x) = 0 \)[/tex]:
[tex]\[ (14 - 2x)(8.5 - 2x)x = 0 \][/tex]
Setting each factor to zero gives:
[tex]\[ x = 0 \][/tex]
[tex]\[ 14 - 2x = 0 \implies x = 7 \][/tex]
[tex]\[ 8.5 - 2x = 0 \implies x = 4.25 \][/tex]
So, the horizontal intercepts are at [tex]\( x = 0 \)[/tex], [tex]\( x = 4.25 \)[/tex], and [tex]\( x = 7 \)[/tex].
2. Vertical intercept (at [tex]\( x = 0 \)[/tex])
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( V(x) \)[/tex]:
[tex]\[ V(0) = (14 - 2 \times 0)(8.5 - 2 \times 0)(0) = 0 \][/tex]
So, the vertical intercept is at [tex]\( (0, 0) \)[/tex].
### Do these points make sense in this situation?
- [tex]\( x = 0 \)[/tex]: This means no squares are cut out, so the box does not exist. Thus, it makes sense to have [tex]\( V(0) = 0 \)[/tex].
- [tex]\( x = 4.25 \)[/tex]: This value is within the range that can physically be cut from the corners (since [tex]\( 4.25 < 8.5/2 = 4.25 \)[/tex] and [tex]\( 4.25 < 14/2 = 7 \)[/tex]). Thus, this makes sense.
- [tex]\( x = 7 \)[/tex]: This value is not really practical because cutting squares of side length 7 inches from an 8.5-inch wide sheet means we have no material left to form the box's height. This value is purely theoretical and results in zero volume because the dimensions intersect and collapse.
### Answer to Question 2:
- Horizontal intercepts: [tex]\( x = 0 \)[/tex], [tex]\( x = 4.25 \)[/tex], and [tex]\( x = 7 \)[/tex]; these points represent the situations where the volume of the box is zero.
- Vertical intercept: At [tex]\( x = 0 \)[/tex], [tex]\( V(0) = 0 \)[/tex], indicating no box can be formed if no material is removed from the corners.
### Given Information
We have a piece of paper that is 8.5 inches by 14 inches. We cut squares of side length [tex]\( x \)[/tex] from each corner and fold up the sides to form a box. The volume [tex]\( V(x) \)[/tex] of the box is given by:
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)x \][/tex]
We need to:
1. Identify the degree and leading term of the polynomial [tex]\( V(x) \)[/tex].
2. Discuss the horizontal and vertical intercepts of the graph of [tex]\( V(x) \)[/tex] and verify if they make sense in this context.
### Question 1: Degree and Leading Term
To identify the degree and leading term:
1. Expand the polynomial [tex]\( V(x) \)[/tex]
[tex]\[ V(x) = (14 - 2x)(8.5 - 2x)x \][/tex]
First, multiply the two binomials:
[tex]\[ (14 - 2x)(8.5 - 2x) \][/tex]
[tex]\[ = 14 \cdot 8.5 - 14 \cdot 2x - 2x \cdot 8.5 + 2x \cdot 2x \][/tex]
[tex]\[ = 119 - 28x - 17x + 4x^2 \][/tex]
[tex]\[ = 119 - 45x + 4x^2 \][/tex]
Now, multiply this result by [tex]\( x \)[/tex]:
[tex]\[ V(x) = (119 - 45x + 4x^2)x \][/tex]
[tex]\[ = 119x - 45x^2 + 4x^3 \][/tex]
2. Identify the degree of the polynomial
The degree is the highest power of [tex]\( x \)[/tex]. Here, the highest power is [tex]\( x^3 \)[/tex], so the degree of the polynomial is 3.
3. Determine the leading term
The leading term is the term with the highest power of [tex]\( x \)[/tex]. In this case, it is [tex]\( 4x^3 \)[/tex].
### Answer to Question 1:
- The degree of the polynomial [tex]\( V(x) \)[/tex] is 3.
- The leading term is [tex]\( 4x^3 \)[/tex].
### Question 2: Horizontal and Vertical Intercepts
1. Horizontal intercepts (where [tex]\( V(x) = 0 \)[/tex])
To find the horizontal intercepts, solve [tex]\( V(x) = 0 \)[/tex]:
[tex]\[ (14 - 2x)(8.5 - 2x)x = 0 \][/tex]
Setting each factor to zero gives:
[tex]\[ x = 0 \][/tex]
[tex]\[ 14 - 2x = 0 \implies x = 7 \][/tex]
[tex]\[ 8.5 - 2x = 0 \implies x = 4.25 \][/tex]
So, the horizontal intercepts are at [tex]\( x = 0 \)[/tex], [tex]\( x = 4.25 \)[/tex], and [tex]\( x = 7 \)[/tex].
2. Vertical intercept (at [tex]\( x = 0 \)[/tex])
Substitute [tex]\( x = 0 \)[/tex] into [tex]\( V(x) \)[/tex]:
[tex]\[ V(0) = (14 - 2 \times 0)(8.5 - 2 \times 0)(0) = 0 \][/tex]
So, the vertical intercept is at [tex]\( (0, 0) \)[/tex].
### Do these points make sense in this situation?
- [tex]\( x = 0 \)[/tex]: This means no squares are cut out, so the box does not exist. Thus, it makes sense to have [tex]\( V(0) = 0 \)[/tex].
- [tex]\( x = 4.25 \)[/tex]: This value is within the range that can physically be cut from the corners (since [tex]\( 4.25 < 8.5/2 = 4.25 \)[/tex] and [tex]\( 4.25 < 14/2 = 7 \)[/tex]). Thus, this makes sense.
- [tex]\( x = 7 \)[/tex]: This value is not really practical because cutting squares of side length 7 inches from an 8.5-inch wide sheet means we have no material left to form the box's height. This value is purely theoretical and results in zero volume because the dimensions intersect and collapse.
### Answer to Question 2:
- Horizontal intercepts: [tex]\( x = 0 \)[/tex], [tex]\( x = 4.25 \)[/tex], and [tex]\( x = 7 \)[/tex]; these points represent the situations where the volume of the box is zero.
- Vertical intercept: At [tex]\( x = 0 \)[/tex], [tex]\( V(0) = 0 \)[/tex], indicating no box can be formed if no material is removed from the corners.
