Answer :
To solve the problem of finding the function representing the height of the shipping box, we have the following information:
1. The volume [tex]\( f(x) \)[/tex] of the rectangular prism is given by:
[tex]\[
f(x) = 2x^3 + 3x^2 - 11x - 6
\][/tex]
2. The length of the shipping box is given by [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x + 3
\][/tex]
3. The width of the shipping box is given by [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = x - 2
\][/tex]
The volume of a rectangular prism is calculated as the product of its length, width, and height. Therefore, we can express this relationship as:
[tex]\[
f(x) = g(x) \cdot h(x) \cdot \text{height}(x)
\][/tex]
To find the function for the height, we need to divide the volume [tex]\( f(x) \)[/tex] by the product of the length [tex]\( g(x) \)[/tex] and the width [tex]\( h(x) \)[/tex]:
[tex]\[
\text{height}(x) = \frac{f(x)}{g(x) \cdot h(x)}
\][/tex]
Substitute the given functions:
[tex]\[
\text{height}(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
When we perform the division, we find that:
[tex]\[
\text{height}(x) = 2x + 1
\][/tex]
Next, let's determine the domain of the height function. The domain includes all real numbers, except where the expression becomes undefined, such as when the denominator is zero. In this case:
- The denominator is [tex]\((x + 3)(x - 2)\)[/tex].
- Setting this equal to zero, we solve:
[tex]\[
(x + 3)(x - 2) = 0
\][/tex]
- This gives [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
Hence, the function is undefined at [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex]. However, since the context of this problem is related to physical dimensions of a box, we must discard [tex]\( x = -3 \)[/tex] because a negative value does not make sense for dimension lengths.
Therefore, the valid domain for the height function, considering it's practically applicable for constructing a box, is [tex]\( x > 2 \)[/tex].
In summary:
- The height function is [tex]\( \text{height}(x) = 2x + 1 \)[/tex].
- The applicable domain for this function is [tex]\( x > 2 \)[/tex].
1. The volume [tex]\( f(x) \)[/tex] of the rectangular prism is given by:
[tex]\[
f(x) = 2x^3 + 3x^2 - 11x - 6
\][/tex]
2. The length of the shipping box is given by [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x + 3
\][/tex]
3. The width of the shipping box is given by [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = x - 2
\][/tex]
The volume of a rectangular prism is calculated as the product of its length, width, and height. Therefore, we can express this relationship as:
[tex]\[
f(x) = g(x) \cdot h(x) \cdot \text{height}(x)
\][/tex]
To find the function for the height, we need to divide the volume [tex]\( f(x) \)[/tex] by the product of the length [tex]\( g(x) \)[/tex] and the width [tex]\( h(x) \)[/tex]:
[tex]\[
\text{height}(x) = \frac{f(x)}{g(x) \cdot h(x)}
\][/tex]
Substitute the given functions:
[tex]\[
\text{height}(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
When we perform the division, we find that:
[tex]\[
\text{height}(x) = 2x + 1
\][/tex]
Next, let's determine the domain of the height function. The domain includes all real numbers, except where the expression becomes undefined, such as when the denominator is zero. In this case:
- The denominator is [tex]\((x + 3)(x - 2)\)[/tex].
- Setting this equal to zero, we solve:
[tex]\[
(x + 3)(x - 2) = 0
\][/tex]
- This gives [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
Hence, the function is undefined at [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex]. However, since the context of this problem is related to physical dimensions of a box, we must discard [tex]\( x = -3 \)[/tex] because a negative value does not make sense for dimension lengths.
Therefore, the valid domain for the height function, considering it's practically applicable for constructing a box, is [tex]\( x > 2 \)[/tex].
In summary:
- The height function is [tex]\( \text{height}(x) = 2x + 1 \)[/tex].
- The applicable domain for this function is [tex]\( x > 2 \)[/tex].
