Question

type the correct answer in the box.
an architect is designing a new building. he makes a model of the building such that the area of the rectangular bas
$12x^2 - 11x - 5$ and the length is $3x + 1$. the completed model will be in the shape of a rectangular prism where the vo
by the polynomial $24x^3 - 58x^2 + 23x + 15$.
determine the width and height of the model in terms of $x$. fill in the values of $m$ and $b$ to complete the expressions
width on the first line and the height on the second line.

Explanation:

Step1: Find the width using area and length

The area of a rectangle is \( A = \text{length} \times \text{width} \). So, width \( w=\frac{\text{Area}}{\text{Length}} \). Given area \( 12x^{2}-11x - 5 \) and length \( 3x + 1 \). We perform polynomial division or factorization. Let's factorize \( 12x^{2}-11x - 5 \):
We need two numbers \( a \) and \( b \) such that \( a\times b=12\times(- 5)=-60 \) and \( a + b=-11 \). The numbers are \( -15 \) and \( 4 \).
\( 12x^{2}-15x + 4x-5=3x(4x - 5)+1(4x - 5)=(3x + 1)(4x - 5) \)
So, width \( w=\frac{(3x + 1)(4x - 5)}{3x + 1}=4x-5 \)

Step2: Find the height using volume, length and width

The volume of a rectangular prism is \( V=\text{length}\times\text{width}\times\text{height} \). So, height \( h = \frac{V}{\text{length}\times\text{width}} \)
We know \( V = 24x^{3}-58x^{2}+23x + 15 \), length \( l=3x + 1 \), width \( w = 4x-5 \)
First, multiply length and width: \( (3x + 1)(4x - 5)=12x^{2}-15x+4x - 5=12x^{2}-11x - 5 \)
Now, divide \( 24x^{3}-58x^{2}+23x + 15 \) by \( 12x^{2}-11x - 5 \)
Using polynomial long division:
Divide \( 24x^{3}\) by \( 12x^{2}\) to get \( 2x \). Multiply \( 12x^{2}-11x - 5 \) by \( 2x \): \( 24x^{3}-22x^{2}-10x \)
Subtract from \( 24x^{3}-58x^{2}+23x + 15 \): \( (24x^{3}-58x^{2}+23x + 15)-(24x^{3}-22x^{2}-10x)=-36x^{2}+33x + 15 \)
Divide \( -36x^{2}\) by \( 12x^{2}\) to get \( - 3 \). Multiply \( 12x^{2}-11x - 5 \) by \( -3 \): \( -36x^{2}+33x + 15 \)
Subtract: \( (-36x^{2}+33x + 15)-(-36x^{2}+33x + 15)=0 \)
So, height \( h = 2x-3 \)

Answer:

Width: \( 4x - 5 \) (so for width in the form \( mx + b \), \( m = 4 \), \( b=-5 \))
Height: \( 2x - 3 \) (so for height in the form \( mx + b \), \( m = 2 \), \( b=-3 \))

(If we assume the expressions for width and height are in the form \( mx + b \), then for width: \( m = 4 \), \( b=-5 \); for height: \( m = 2 \), \( b=-3 \))