Question

Please help me with this is so confusing

Answer 1

Answer:

The expression for the height of the solid is:

$\displaystyle h = x^2+x-9$

Step-by-step explanation:

Recall that the volume of a rectangular solid is given by:

$\displaystyle V = \ell wh$

Where l is the length, w is the width, and h is the height.

We know that the volume is given by the polynomial:

$\displaystyle V = 3x^4-3x^3-33x^2+54x$

And that the length and width are given by, respectively:

$\displaystyle \ell = 3x \text{ and } w =x-2$

Substitute:

$\displaystyle 3x^4-3x^3-33x^2+54x=(3x)(x-2)h$

We can solve for h. First, divide both sides by 3x:

$\displaystyle \frac{3x^4-3x^3-33x^2+54x}{3x}=(x-2)h$

Divide each term:

$\displaystyle x^3-x^2-11x+18=(x-2)h$

To solve for h, divide both sides by (x - 2):

$\displaystyle h = \frac{x^3-x^2-11x+18}{x-2}$

Since this is a polynomial divided by a binomial in the form of (x - a), we can use synthetic division, where a = 2. This is shown below. Therefore, the expression for the height of the solid is:

$\displaystyle h = x^2+x-9$