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Answer :
To find the height of a hexagonal prism, we need to use the relationship between the volume of the prism, the area of the base, and the height. The volume [tex]\( V \)[/tex] of a prism is given by the formula:
[tex]\[ V = \text{Base Area} \times \text{Height} \][/tex]
In this case, we're given:
- The volume of the hexagonal prism as [tex]\( 35x^5 + 20x^4 - 57x^3 + x^2 + 18x - 30 \)[/tex].
- The area of the base as [tex]\( 5x^2 - 6 \)[/tex].
To find the height, we need to divide the volume by the area of the base:
[tex]\[ \text{Height} = \frac{\text{Volume}}{\text{Base Area}} \][/tex]
Plugging in the given expressions:
[tex]\[ \text{Height} = \frac{35x^5 + 20x^4 - 57x^3 + x^2 + 18x - 30}{5x^2 - 6} \][/tex]
Performing the division gives us the following expression for the height:
[tex]\[ \text{Height} = 7x^3 + 4x^2 - 3x + 5 \][/tex]
This means that the height of the hexagonal prism, as a function of [tex]\( x \)[/tex], is [tex]\( 7x^3 + 4x^2 - 3x + 5 \)[/tex].
[tex]\[ V = \text{Base Area} \times \text{Height} \][/tex]
In this case, we're given:
- The volume of the hexagonal prism as [tex]\( 35x^5 + 20x^4 - 57x^3 + x^2 + 18x - 30 \)[/tex].
- The area of the base as [tex]\( 5x^2 - 6 \)[/tex].
To find the height, we need to divide the volume by the area of the base:
[tex]\[ \text{Height} = \frac{\text{Volume}}{\text{Base Area}} \][/tex]
Plugging in the given expressions:
[tex]\[ \text{Height} = \frac{35x^5 + 20x^4 - 57x^3 + x^2 + 18x - 30}{5x^2 - 6} \][/tex]
Performing the division gives us the following expression for the height:
[tex]\[ \text{Height} = 7x^3 + 4x^2 - 3x + 5 \][/tex]
This means that the height of the hexagonal prism, as a function of [tex]\( x \)[/tex], is [tex]\( 7x^3 + 4x^2 - 3x + 5 \)[/tex].
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