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To determine the width of the driveway and the height of the carport, we need to follow a series of steps involving algebraic manipulation.
### Width of the Entire Driveway
1. Given:
- Area of the driveway: [tex]\(5x^2 + 43x - 18\)[/tex]
- Length of the driveway: [tex]\(x + 9\)[/tex]
2. Formula:
- The width of the driveway is given by [tex]\(\frac{\text{Area}}{\text{Length}}\)[/tex].
3. Calculation:
- Divide the polynomial for the area by the polynomial for the length.
[tex]\[ \frac{5x^2 + 43x - 18}{x + 9} \][/tex]
To perform this division, we use polynomial long division.
- Step 1: Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{5x^2}{x} = 5x \][/tex]
- Step 2: Multiply [tex]\(5x\)[/tex] by [tex]\(x + 9\)[/tex]:
[tex]\[ 5x \cdot (x + 9) = 5x^2 + 45x \][/tex]
- Step 3: Subtract this product from the original polynomial:
[tex]\[ (5x^2 + 43x - 18) - (5x^2 + 45x) = -2x - 18 \][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
- Step 5: Multiply [tex]\(-2\)[/tex] by the divisor:
[tex]\[ -2 \cdot (x + 9) = -2x - 18 \][/tex]
- Step 6: Subtract this product from [tex]\(-2x - 18\)[/tex]:
[tex]\[ (-2x - 18) - (-2x - 18) = 0 \][/tex]
The division is exact, and no remainder is present.
Therefore, the width of the driveway is [tex]\(5x - 2\)[/tex].
### Height of the Carport
1. Given:
- Volume of the carport: [tex]\(48x^3 + 68x^2 - 8x - 3\)[/tex]
- Area under the carport: [tex]\(8x^2 + 10x - 3\)[/tex]
2. Formula:
- The height of the carport is given by [tex]\(\frac{\text{Volume}}{\text{Area}}\)[/tex].
3. Calculation:
- Divide the polynomial for the volume by the polynomial for the area.
[tex]\[ \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3} \][/tex]
To perform this division, again use polynomial long division.
- Step 1: Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{48x^3}{8x^2} = 6x \][/tex]
- Step 2: Multiply [tex]\(6x\)[/tex] by [tex]\(8x^2 + 10x - 3\)[/tex]:
[tex]\[ 6x \cdot (8x^2 + 10x - 3) = 48x^3 + 60x^2 - 18x \][/tex]
- Step 3: Subtract this product from the original polynomial:
[tex]\[ (48x^3 + 68x^2 - 8x - 3) - (48x^3 + 60x^2 - 18x) = 8x^2 + 10x - 3 \][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{8x^2}{8x^2} = 1 \][/tex]
- Step 5: Multiply [tex]\(1\)[/tex] by the divisor:
[tex]\[ 1 \cdot (8x^2 + 10x - 3) = 8x^2 + 10x - 3 \][/tex]
- Step 6: Subtract this product from [tex]\(8x^2 + 10x - 3\)[/tex]:
[tex]\[ (8x^2 + 10x - 3) - (8x^2 + 10x - 3) = 0 \][/tex]
The division is exact, and no remainder is present.
Therefore, the height of the carport is [tex]\(6x + 1\)[/tex].
### Final Expressions
So, the width of the entire driveway is:
[tex]\[ 5x - 2 \][/tex]
And the height of the carport is:
[tex]\[ 6x + 1 \][/tex]
### Width of the Entire Driveway
1. Given:
- Area of the driveway: [tex]\(5x^2 + 43x - 18\)[/tex]
- Length of the driveway: [tex]\(x + 9\)[/tex]
2. Formula:
- The width of the driveway is given by [tex]\(\frac{\text{Area}}{\text{Length}}\)[/tex].
3. Calculation:
- Divide the polynomial for the area by the polynomial for the length.
[tex]\[ \frac{5x^2 + 43x - 18}{x + 9} \][/tex]
To perform this division, we use polynomial long division.
- Step 1: Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{5x^2}{x} = 5x \][/tex]
- Step 2: Multiply [tex]\(5x\)[/tex] by [tex]\(x + 9\)[/tex]:
[tex]\[ 5x \cdot (x + 9) = 5x^2 + 45x \][/tex]
- Step 3: Subtract this product from the original polynomial:
[tex]\[ (5x^2 + 43x - 18) - (5x^2 + 45x) = -2x - 18 \][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{-2x}{x} = -2 \][/tex]
- Step 5: Multiply [tex]\(-2\)[/tex] by the divisor:
[tex]\[ -2 \cdot (x + 9) = -2x - 18 \][/tex]
- Step 6: Subtract this product from [tex]\(-2x - 18\)[/tex]:
[tex]\[ (-2x - 18) - (-2x - 18) = 0 \][/tex]
The division is exact, and no remainder is present.
Therefore, the width of the driveway is [tex]\(5x - 2\)[/tex].
### Height of the Carport
1. Given:
- Volume of the carport: [tex]\(48x^3 + 68x^2 - 8x - 3\)[/tex]
- Area under the carport: [tex]\(8x^2 + 10x - 3\)[/tex]
2. Formula:
- The height of the carport is given by [tex]\(\frac{\text{Volume}}{\text{Area}}\)[/tex].
3. Calculation:
- Divide the polynomial for the volume by the polynomial for the area.
[tex]\[ \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3} \][/tex]
To perform this division, again use polynomial long division.
- Step 1: Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[ \frac{48x^3}{8x^2} = 6x \][/tex]
- Step 2: Multiply [tex]\(6x\)[/tex] by [tex]\(8x^2 + 10x - 3\)[/tex]:
[tex]\[ 6x \cdot (8x^2 + 10x - 3) = 48x^3 + 60x^2 - 18x \][/tex]
- Step 3: Subtract this product from the original polynomial:
[tex]\[ (48x^3 + 68x^2 - 8x - 3) - (48x^3 + 60x^2 - 18x) = 8x^2 + 10x - 3 \][/tex]
- Step 4: Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{8x^2}{8x^2} = 1 \][/tex]
- Step 5: Multiply [tex]\(1\)[/tex] by the divisor:
[tex]\[ 1 \cdot (8x^2 + 10x - 3) = 8x^2 + 10x - 3 \][/tex]
- Step 6: Subtract this product from [tex]\(8x^2 + 10x - 3\)[/tex]:
[tex]\[ (8x^2 + 10x - 3) - (8x^2 + 10x - 3) = 0 \][/tex]
The division is exact, and no remainder is present.
Therefore, the height of the carport is [tex]\(6x + 1\)[/tex].
### Final Expressions
So, the width of the entire driveway is:
[tex]\[ 5x - 2 \][/tex]
And the height of the carport is:
[tex]\[ 6x + 1 \][/tex]
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