Question

type the correct answer in the box.
sharon is paving a rectangular concrete driveway on the side of her house. the area of the driveway is $5x^2 + 43x - 18$, and the length of the driveway is $x + 9$.
additionally, sharon plans to install a carport over a small portion of the driveway. the volume that the carport can enclose is $48x^3 + 68x^2 - 8x - 3$, and the area of driveway beneath the carport is $8x^2 + 10x - 3$
determine the width of the entire driveway and height of the carport in terms of $x$. replace the values of $m$ and $b$ to complete the expression that represents the width of the entire driveway on the first line, and then replace the values of $m$ and $b$ to complete the expression that represents the height of the carport on the second line.
width: $mx + b$
height: $mx + b$

Explanation:

Step1: Factor area for driveway width

We factor the quadratic area $5x^2 + 43x - 18$ by dividing by the length $x+9$. Using polynomial long division or factoring:
$5x^2 + 43x - 18 = (x+9)(5x - 2)$

Step2: Identify driveway width

Since $\text{Area} = \text{Length} \times \text{Width}$, we divide area by length:
$\text{Width} = \frac{5x^2 + 43x - 18}{x+9} = 5x - 2$

Step3: Factor volume for carport height

We factor the cubic volume $48x^3 + 68x^2 - 8x - 3$ by dividing by the base area $8x^2 + 10x - 3$. Using polynomial long division or factoring:
$48x^3 + 68x^2 - 8x - 3 = (8x^2 + 10x - 3)(6x + 1)$

Step4: Identify carport height

Since $\text{Volume} = \text{Base Area} \times \text{Height}$, we divide volume by base area:
$\text{Height} = \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3} = 6x + 1$

Answer:

Width: $5x - 2$
Height: $6x + 1$