Answer :
To solve this problem, we need to find the function that represents the height of the shipping box and determine its domain. Here's a step-by-step explanation:
1. Understanding the Functions:
- The volume of the shipping box, which is a rectangular prism, is given by the function [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex].
- The length of the box is given by the function [tex]\( g(x) = x + 3 \)[/tex].
- The width of the box is given by the function [tex]\( h(x) = x - 2 \)[/tex].
2. Finding the Height Function:
- Since the volume of a rectangular prism is calculated using the formula Volume = Length [tex]\(\times\)[/tex] Width [tex]\(\times\)[/tex] Height, we can express the height in terms of [tex]\( x \)[/tex] by dividing the volume function [tex]\( f(x) \)[/tex] by the product of the length and width functions [tex]\( g(x) \times h(x) \)[/tex].
- So, the height function [tex]\( H(x) \)[/tex] is given by:
[tex]\[
H(x) = \frac{f(x)}{g(x) \cdot h(x)}
\][/tex]
- Plugging in the given functions, we get:
[tex]\[
H(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
3. Simplifying the Height Function:
- When we simplify the fraction [tex]\(\frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}\)[/tex], we find that the simplified form of the height function is:
[tex]\[
H(x) = 2x + 1
\][/tex]
4. Determining the Domain:
- The domain of the height function is determined by identifying the values of [tex]\( x \)[/tex] that make the denominator zero, as the division by zero is undefined.
- From [tex]\( (x + 3)(x - 2) \)[/tex], we find that it is zero when [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
- Therefore, the values [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex] must be excluded from the domain.
- Consequently, the domain of the function is all real numbers except [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
In conclusion, the function that represents the height of the shipping box is [tex]\( H(x) = 2x + 1 \)[/tex], and its domain is all real numbers except [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
1. Understanding the Functions:
- The volume of the shipping box, which is a rectangular prism, is given by the function [tex]\( f(x) = 2x^3 + 3x^2 - 11x - 6 \)[/tex].
- The length of the box is given by the function [tex]\( g(x) = x + 3 \)[/tex].
- The width of the box is given by the function [tex]\( h(x) = x - 2 \)[/tex].
2. Finding the Height Function:
- Since the volume of a rectangular prism is calculated using the formula Volume = Length [tex]\(\times\)[/tex] Width [tex]\(\times\)[/tex] Height, we can express the height in terms of [tex]\( x \)[/tex] by dividing the volume function [tex]\( f(x) \)[/tex] by the product of the length and width functions [tex]\( g(x) \times h(x) \)[/tex].
- So, the height function [tex]\( H(x) \)[/tex] is given by:
[tex]\[
H(x) = \frac{f(x)}{g(x) \cdot h(x)}
\][/tex]
- Plugging in the given functions, we get:
[tex]\[
H(x) = \frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}
\][/tex]
3. Simplifying the Height Function:
- When we simplify the fraction [tex]\(\frac{2x^3 + 3x^2 - 11x - 6}{(x + 3)(x - 2)}\)[/tex], we find that the simplified form of the height function is:
[tex]\[
H(x) = 2x + 1
\][/tex]
4. Determining the Domain:
- The domain of the height function is determined by identifying the values of [tex]\( x \)[/tex] that make the denominator zero, as the division by zero is undefined.
- From [tex]\( (x + 3)(x - 2) \)[/tex], we find that it is zero when [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
- Therefore, the values [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex] must be excluded from the domain.
- Consequently, the domain of the function is all real numbers except [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
In conclusion, the function that represents the height of the shipping box is [tex]\( H(x) = 2x + 1 \)[/tex], and its domain is all real numbers except [tex]\( x = -3 \)[/tex] and [tex]\( x = 2 \)[/tex].
