Question

You have a rectangular space where you plan to create an obstacle course for an animal. The area of the rectangular space is represented by the expression 6×2 − 16x. The width of the rectangular space is represented by the expression 2x.

Part A: Write an expression to represent the length of the rectangular space. Then simplify your expression. Show all your work. (6 points)

Part B: Prove that your answer in part A is correct by multiplying the length and the width of the rectangle. Show all your work. (4 points)

1. Part A:
To find the length of the rectangular space, we need to use the formula for the area of a rectangle, which is length * width. We are given that the area is represented by the expression 6x^2 – 16x and the width is represented by the expression 2x.
We can set up the following equation:
length * width = 6x^2 – 16x
length = (6x^2 – 16x) / width
Substituting the expression for width, we get:
length = (6x^2 – 16x) / (2x)
Simplifying the expression gives us:
length = 3x – 8
Part B:
To prove that our expression for the length is correct, we can multiply it by the width to obtain the area of the rectangle.
length * width = (3x – 8) * (2x)
length * width = 6x^2 – 16x
This is the same expression that we were given for the area of the rectangle, so our expression for the length is correct.