Question

1. if this rectangle has a perimeter of 84, then what is the area? x - 7 3x + 1

Explanation:

Step1: Recall perimeter formula for rectangle

The perimeter formula of a rectangle is $P = 2(l + w)$, where $l$ is the length and $w$ is the width. Here, $l=3x + 1$ and $w=x - 7$, and $P = 84$. So, $84=2((3x + 1)+(x - 7))$.

Step2: Simplify the equation

First, simplify the expression inside the parentheses: $(3x + 1)+(x - 7)=3x+1+x - 7=4x - 6$. Then the equation becomes $84 = 2(4x - 6)$. Distribute the 2: $84=8x-12$.

Step3: Solve for x

Add 12 to both sides of the equation: $84 + 12=8x-12 + 12$, which gives $96 = 8x$. Divide both sides by 8: $x=\frac{96}{8}=12$.

Step4: Find the length and width

Substitute $x = 12$ into the expressions for length and width. Length $l=3x + 1=3\times12 + 1=36 + 1=37$. Width $w=x - 7=12 - 7 = 5$.

Step5: Calculate the area

The area formula of a rectangle is $A=l\times w$. So, $A=37\times5 = 185$.

Answer:

185