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Sharon is paving a rectangular concrete driveway on the side of her house. The area of the driveway is [tex]5x^2 + 43x - 18[/tex], and the length of the driveway is [tex]x + 9[/tex].

Additionally, Sharon plans to install a carport over a small portion of the driveway. The volume that the carport can enclose is [tex]48x^3 + 68x^2 - 8x - 3[/tex], and the area of the driveway beneath the carport is [tex]8x^2 + 10x - 3[/tex].

Determine the width of the entire driveway and the height of the carport in terms of [tex]x[/tex]. Replace the values of [tex]m[/tex] and [tex]b[/tex] to complete the expression that represents the width of the entire driveway on the first line, and then replace the values of [tex]m[/tex] and [tex]b[/tex] to complete the expression that represents the height of the carport on the second line.

Width: [tex]mx + b[/tex]

Height: [tex]mx + b[/tex]


Sagot :

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]