College

The volume of a hexagonal prism is given by [tex]$35x^5 + 20x^4 - 57x^3 + x^2 + 18x - 30$[/tex]. The area of the base is given by [tex]$5x^2 - 6$[/tex]. Find an expression for the height of the hexagonal prism.

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].