Question

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: Find driveway width

Width = $\frac{\text{Area}}{\text{Length}}$
$\text{Width} = \frac{5x^2 + 43x - 18}{x + 9}$
Use polynomial division or factoring:
$5x^2 + 43x - 18 = (x + 9)(5x - 2)$
So $\text{Width} = 5x - 2$

Step2: Find carport height

Height = $\frac{\text{Volume}}{\text{Area beneath carport}}$
$\text{Height} = \frac{48x^3 + 68x^2 - 8x - 3}{8x^2 + 10x - 3}$
Use polynomial division or factoring:
$48x^3 + 68x^2 - 8x - 3 = (8x^2 + 10x - 3)(6x + 1)$
So $\text{Height} = 6x + 1$

Answer:

Width of the entire driveway: $5x - 2$ (so $m=5$, $b=-2$)
Height of the carport: $6x + 1$ (so $m=6$, $b=1$)