Answer :
To solve this question, we'll need to determine two functions: the width of the driveway and the height of the carport, both in terms of [tex]\( x \)[/tex]. We'll proceed step-by-step.
### Width of the Driveway
The area of the driveway, [tex]\( A \)[/tex], and the length of the driveway, [tex]\( L \)[/tex], are given by:
[tex]\[ A = 5x^2 + 43x - 18 \][/tex]
[tex]\[ L = x + 9 \][/tex]
The width of the driveway, [tex]\( W \)[/tex], can be determined by dividing the area by the length:
[tex]\[ W = \frac{A}{L} = \frac{5x^2 + 43x - 18}{x + 9} \][/tex]
We need to perform polynomial long division to simplify this fraction.
#### Polynomial Division
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[ \frac{5x^2}{x} = 5x \][/tex]
2. Multiply [tex]\( x + 9 \)[/tex] by [tex]\( 5x \)[/tex]:
[tex]\[ (x + 9) \cdot 5x = 5x^2 + 45x \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (5x^2 + 43x - 18) - (5x^2 + 45x) = 43x - 45x - 18 = -2x - 18 \][/tex]
4. Divide the first term of the new numerator by the first term of the denominator:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
5. Multiply [tex]\( x + 9 \)[/tex] by [tex]\( -2 \)[/tex]:
[tex]\[ (x + 9) \cdot -2 = -2x - 18 \][/tex]
6. Subtract this product from the current numerator:
[tex]\[ (-2x - 18) - (-2x - 18) = 0 \][/tex]
So, the width [tex]\( W \)[/tex] is:
[tex]\[ W = 5x - 2 \][/tex]
Therefore, [tex]\( m = 5 \)[/tex] and [tex]\( b = -2 \)[/tex].
### Height of the Carport
The volume of the carport, [tex]\( V \)[/tex], and the area beneath the carport, [tex]\( A_{\text{carport}} \)[/tex], are given by:
[tex]\[ V = 48x^3 + 68x^2 - 8x - 3 \][/tex]
[tex]\[ A_{\text{carport}} = 8x^2 + 10x - 3 \][/tex]
The height of the carport, [tex]\( H \)[/tex], can be determined by dividing the volume by the area:
[tex]\[ H = \frac{V}{A_{\text{carport}}} = \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3} \][/tex]
We need to perform polynomial long division to simplify this fraction.
#### Polynomial Division
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[ \frac{48x^3}{8x^2} = 6x \][/tex]
2. Multiply [tex]\( 8x^2 + 10x - 3 \)[/tex] by [tex]\( 6x \)[/tex]:
[tex]\[ (8x^2 + 10x - 3) \cdot 6x = 48x^3 + 60x^2 - 18x \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (48x^3 + 68x^2 - 8x - 3) - (48x^3 + 60x^2 - 18x) = 68x^2 - 60x^2 + 10x - 3 = 8x^2 + 10x - 3 \][/tex]
4. Divide the first term of the new numerator by the first term of the denominator:
[tex]\[ \frac{8x^2}{8x^2} = 1 \][/tex]
5. Multiply [tex]\( 8x^2 + 10x - 3 \)[/tex] by [tex]\( 1 \)[/tex]:
[tex]\[ (8x^2 + 10x - 3) \cdot 1 = 8x^2 + 10x - 3 \][/tex]
6. Subtract this product from the current numerator:
[tex]\[ (8x^2 + 10x - 3) - (8x^2 + 10x - 3) = 0 \][/tex]
So, the height [tex]\( H \)[/tex] is:
[tex]\[ H = 6x + 1 \][/tex]
Therefore, [tex]\( m = 6 \)[/tex] and [tex]\( b = 1 \)[/tex].
### Final Answer
Width: [tex]\( 5x - 2 \)[/tex]
Height: [tex]\( 6x + 1 \)[/tex]
Replace the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
Width: [tex]\( \boxed{5} x + \boxed{-2} \)[/tex]
Height: [tex]\( \boxed{6} x + \boxed{1} \)[/tex]
### Width of the Driveway
The area of the driveway, [tex]\( A \)[/tex], and the length of the driveway, [tex]\( L \)[/tex], are given by:
[tex]\[ A = 5x^2 + 43x - 18 \][/tex]
[tex]\[ L = x + 9 \][/tex]
The width of the driveway, [tex]\( W \)[/tex], can be determined by dividing the area by the length:
[tex]\[ W = \frac{A}{L} = \frac{5x^2 + 43x - 18}{x + 9} \][/tex]
We need to perform polynomial long division to simplify this fraction.
#### Polynomial Division
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[ \frac{5x^2}{x} = 5x \][/tex]
2. Multiply [tex]\( x + 9 \)[/tex] by [tex]\( 5x \)[/tex]:
[tex]\[ (x + 9) \cdot 5x = 5x^2 + 45x \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (5x^2 + 43x - 18) - (5x^2 + 45x) = 43x - 45x - 18 = -2x - 18 \][/tex]
4. Divide the first term of the new numerator by the first term of the denominator:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
5. Multiply [tex]\( x + 9 \)[/tex] by [tex]\( -2 \)[/tex]:
[tex]\[ (x + 9) \cdot -2 = -2x - 18 \][/tex]
6. Subtract this product from the current numerator:
[tex]\[ (-2x - 18) - (-2x - 18) = 0 \][/tex]
So, the width [tex]\( W \)[/tex] is:
[tex]\[ W = 5x - 2 \][/tex]
Therefore, [tex]\( m = 5 \)[/tex] and [tex]\( b = -2 \)[/tex].
### Height of the Carport
The volume of the carport, [tex]\( V \)[/tex], and the area beneath the carport, [tex]\( A_{\text{carport}} \)[/tex], are given by:
[tex]\[ V = 48x^3 + 68x^2 - 8x - 3 \][/tex]
[tex]\[ A_{\text{carport}} = 8x^2 + 10x - 3 \][/tex]
The height of the carport, [tex]\( H \)[/tex], can be determined by dividing the volume by the area:
[tex]\[ H = \frac{V}{A_{\text{carport}}} = \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3} \][/tex]
We need to perform polynomial long division to simplify this fraction.
#### Polynomial Division
1. Divide the first term of the numerator by the first term of the denominator:
[tex]\[ \frac{48x^3}{8x^2} = 6x \][/tex]
2. Multiply [tex]\( 8x^2 + 10x - 3 \)[/tex] by [tex]\( 6x \)[/tex]:
[tex]\[ (8x^2 + 10x - 3) \cdot 6x = 48x^3 + 60x^2 - 18x \][/tex]
3. Subtract this product from the original numerator:
[tex]\[ (48x^3 + 68x^2 - 8x - 3) - (48x^3 + 60x^2 - 18x) = 68x^2 - 60x^2 + 10x - 3 = 8x^2 + 10x - 3 \][/tex]
4. Divide the first term of the new numerator by the first term of the denominator:
[tex]\[ \frac{8x^2}{8x^2} = 1 \][/tex]
5. Multiply [tex]\( 8x^2 + 10x - 3 \)[/tex] by [tex]\( 1 \)[/tex]:
[tex]\[ (8x^2 + 10x - 3) \cdot 1 = 8x^2 + 10x - 3 \][/tex]
6. Subtract this product from the current numerator:
[tex]\[ (8x^2 + 10x - 3) - (8x^2 + 10x - 3) = 0 \][/tex]
So, the height [tex]\( H \)[/tex] is:
[tex]\[ H = 6x + 1 \][/tex]
Therefore, [tex]\( m = 6 \)[/tex] and [tex]\( b = 1 \)[/tex].
### Final Answer
Width: [tex]\( 5x - 2 \)[/tex]
Height: [tex]\( 6x + 1 \)[/tex]
Replace the values of [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
Width: [tex]\( \boxed{5} x + \boxed{-2} \)[/tex]
Height: [tex]\( \boxed{6} x + \boxed{1} \)[/tex]
