Question

the length of a rectangle is six times its width. if the area of the rectangle is 384 yd², find its perimeter.

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) yards. Then the length \( l \) is \( 6w \) yards (since length is six times the width).

Step2: Use the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = l \times w \). We know the area is \( 384 \, \text{yd}^2 \), so substitute \( l = 6w \) into the area formula:
\[
384 = 6w \times w
\]
Simplify the right - hand side:
\[
384 = 6w^{2}
\]

Step3: Solve for \( w \)

Divide both sides of the equation \( 6w^{2}=384 \) by 6:
\[
w^{2}=\frac{384}{6}=64
\]
Take the square root of both sides. Since width cannot be negative, we have \( w = \sqrt{64}=8 \) yards.

Step4: Find the length

Now that we know \( w = 8 \) yards, the length \( l=6w = 6\times8 = 48 \) yards.

Step5: Calculate the perimeter

The perimeter \( P \) of a rectangle is given by the formula \( P = 2(l + w) \). Substitute \( l = 48 \) and \( w = 8 \) into the formula:
\[
P=2(48 + 8)=2\times56 = 112
\]

Answer:

\( 112 \)