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 10x² - 6x. 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)

Explanation:

Step1: Recall area formula

The area of a rectangle is $A = l\times w$, where $A$ is area, $l$ is length and $w$ is width. We know $A = 10x^{2}-6x$ and $w = 2x$, and we need to find $l$. So, $l=\frac{A}{w}$.

Step2: Substitute values and simplify

$l=\frac{10x^{2}-6x}{2x}=\frac{10x^{2}}{2x}-\frac{6x}{2x}$.
For $\frac{10x^{2}}{2x}$, using the rule of exponents $\frac{a^{m}}{a^{n}}=a^{m - n}$, we have $\frac{10}{2}x^{2 - 1}=5x$.
For $\frac{6x}{2x}$, $\frac{6}{2}x^{1 - 1}=3$. So, $l = 5x-3$.

Step3: Prove the answer in part A

Multiply the length $l = 5x - 3$ and the width $w=2x$.
Using the distributive property $a(b + c)=ab+ac$, we have $(5x - 3)\times2x=5x\times2x-3\times2x$.
$5x\times2x = 10x^{2}$ and $3\times2x=6x$. So, $(5x - 3)\times2x=10x^{2}-6x$, which is the given area.

Answer:

Part A: The expression for the length is $\frac{10x^{2}-6x}{2x}$, and the simplified expression is $5x - 3$.
Part B: $(5x - 3)\times2x=5x\times2x-3\times2x=10x^{2}-6x$, which proves the length expression in part A is correct.