Question

the length of a rectangle is six times its width. if the perimeter of the rectangle is 84 m, find its area.

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) meters. Then the length \( l \) is \( 6w \) meters.

Step2: Use perimeter formula

The perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \). We know \( P = 84 \) m, so substitute \( l = 6w \) into the formula:
\[
84 = 2(6w + w)
\]

Step3: Simplify and solve for \( w \)

Simplify the equation inside the parentheses:
\[
84 = 2(7w)
\]
\[
84 = 14w
\]
Divide both sides by 14:
\[
w = \frac{84}{14} = 6
\]

Step4: Find the length

Since \( l = 6w \), substitute \( w = 6 \):
\[
l = 6 \times 6 = 36
\]

Step5: Calculate the area

The area \( A \) of a rectangle is \( A = l \times w \). Substitute \( l = 36 \) and \( w = 6 \):
\[
A = 36 \times 6 = 216
\]

Answer:

The area of the rectangle is \( 216 \) square meters.