Answer :
To find the height of the shipping box, we need to understand the relationships between the volume, length, width, and height of a rectangular prism.
### Given:
- Volume of the box: [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex]
- Length of the box: [tex]\( g(x) = x + 3 \)[/tex]
- Width of the box: [tex]\( h(x) = x - 2 \)[/tex]
### Objective:
We want to determine a function that represents the height of the box.
### Relationship:
The formula for the volume of a rectangular prism is given by:
[tex]\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\][/tex]
Which can be rewritten in terms of functions:
[tex]\[
f(x) = g(x) \times h(x) \times \text{Height}(x)
\][/tex]
### Finding the Height:
To find the expression for the height function, we can rearrange the formula to solve for [tex]\(\text{Height}(x)\)[/tex]:
[tex]\[
\text{Height}(x) = \frac{f(x)}{g(x) \times h(x)}
\][/tex]
### Calculation:
1. Volume function: [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex]
2. Length function: [tex]\( g(x) = x + 3 \)[/tex]
3. Width function: [tex]\( h(x) = x - 2 \)[/tex]
Now compute:
[tex]\[
g(x) \times h(x) = (x + 3)(x - 2)
\][/tex]
Using the relationship:
[tex]\[
\text{Height}(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
Substituting the expression for [tex]\(\text{Height}(x)\)[/tex]:
[tex]\[
\text{Height}(x) = 2x + 1
\][/tex]
### Domain:
For the height function to be defined, the denominator [tex]\((x + 3)(x - 2)\)[/tex] must not be zero:
1. Set [tex]\( (x + 3)(x - 2) = 0 \)[/tex] to find points where the denominator is zero.
2. Solve: [tex]\( x + 3 = 0 \)[/tex] gives [tex]\( x = -3 \)[/tex]
3. Solve: [tex]\( x - 2 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex]
These are the points where the function is undefined. Thus, the domain of the height function excludes these values:
- [tex]\(-\infty < x < -3\)[/tex] and [tex]\(2 < x < \infty\)[/tex]
So, the height function is:
[tex]\[
\text{Height}(x) = 2x + 1
\][/tex]
And the domain is:
[tex]\[
x \in (-\infty, -3) \cup (2, \infty)
\][/tex]
This gives us the complete solution for the height of the shipping box.
### Given:
- Volume of the box: [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex]
- Length of the box: [tex]\( g(x) = x + 3 \)[/tex]
- Width of the box: [tex]\( h(x) = x - 2 \)[/tex]
### Objective:
We want to determine a function that represents the height of the box.
### Relationship:
The formula for the volume of a rectangular prism is given by:
[tex]\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\][/tex]
Which can be rewritten in terms of functions:
[tex]\[
f(x) = g(x) \times h(x) \times \text{Height}(x)
\][/tex]
### Finding the Height:
To find the expression for the height function, we can rearrange the formula to solve for [tex]\(\text{Height}(x)\)[/tex]:
[tex]\[
\text{Height}(x) = \frac{f(x)}{g(x) \times h(x)}
\][/tex]
### Calculation:
1. Volume function: [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex]
2. Length function: [tex]\( g(x) = x + 3 \)[/tex]
3. Width function: [tex]\( h(x) = x - 2 \)[/tex]
Now compute:
[tex]\[
g(x) \times h(x) = (x + 3)(x - 2)
\][/tex]
Using the relationship:
[tex]\[
\text{Height}(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
Substituting the expression for [tex]\(\text{Height}(x)\)[/tex]:
[tex]\[
\text{Height}(x) = 2x + 1
\][/tex]
### Domain:
For the height function to be defined, the denominator [tex]\((x + 3)(x - 2)\)[/tex] must not be zero:
1. Set [tex]\( (x + 3)(x - 2) = 0 \)[/tex] to find points where the denominator is zero.
2. Solve: [tex]\( x + 3 = 0 \)[/tex] gives [tex]\( x = -3 \)[/tex]
3. Solve: [tex]\( x - 2 = 0 \)[/tex] gives [tex]\( x = 2 \)[/tex]
These are the points where the function is undefined. Thus, the domain of the height function excludes these values:
- [tex]\(-\infty < x < -3\)[/tex] and [tex]\(2 < x < \infty\)[/tex]
So, the height function is:
[tex]\[
\text{Height}(x) = 2x + 1
\][/tex]
And the domain is:
[tex]\[
x \in (-\infty, -3) \cup (2, \infty)
\][/tex]
This gives us the complete solution for the height of the shipping box.
