The volume of a rectangular prism is given by the expression [tex]10x^3 + 46x^2 - 21x - 27[/tex]. The area of the base of the prism is given by the expression [tex]2x^2 + 8x - 9[/tex]. Which of the following expressions represents the height of the prism? (Recall that [tex]V = B \cdot h[/tex].)

A. [tex]8x - 3[/tex]
B. [tex]3x - 5[/tex]
C. [tex]5x + 3[/tex]
D. [tex]42x + 3[/tex]



Answer :

To find the height of a rectangular prism given its volume and base area, we use the formula for volume which is [tex]\( V = B \times h \)[/tex], where [tex]\( V \)[/tex] is the volume, [tex]\( B \)[/tex] is the area of the base, and [tex]\( h \)[/tex] is the height. Given are the expressions for [tex]\( V \)[/tex] and [tex]\( B \)[/tex]:

[tex]\[ V = 10x^3 + 46x^2 - 21x - 27 \][/tex]
[tex]\[ B = 2x^2 + 8x - 9 \][/tex]

We need to determine the height expression, [tex]\( h \)[/tex], which can be found by rearranging the formula to solve for [tex]\( h \)[/tex]:

[tex]\[ h = \frac{V}{B} \][/tex]

Substituting the given expressions:

[tex]\[ h = \frac{10x^3 + 46x^2 - 21x - 27}{2x^2 + 8x - 9} \][/tex]

We simplify this rational expression by performing polynomial division or factorization. The resulting expression for the height [tex]\( h \)[/tex] simplifies to:

[tex]\[ h = 5x + 3 \][/tex]

Thus, the expression that represents the height of the prism is:

[tex]\[ 5x + 3 \][/tex]

So out of the given choices, the correct answer is:

[tex]\[ \boxed{5x + 3} \][/tex]