Question

the area of a rectangle is 52 yd², and the length of the rectangle is 5 yd less than twice the width. find the dimensions of the rectangle. length : □ yd width : □ yd

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) yards. Then the length \( l \) is \( 2w - 5 \) yards (since length is 5 yd less than twice the width).

Step2: Use area formula

The area of a rectangle is \( A = l \times w \). We know \( A = 52 \) yd², so substitute \( l = 2w - 5 \) into the formula:
\[
(2w - 5)w = 52
\]
Expand the left side:
\[
2w^2 - 5w = 52
\]
Rearrange to form a quadratic equation:
\[
2w^2 - 5w - 52 = 0
\]

Step3: Solve the quadratic equation

We can solve \( 2w^2 - 5w - 52 = 0 \) using the quadratic formula \( w=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -5 \), and \( c = -52 \).

First, calculate the discriminant \( D = b^2 - 4ac \):
\[
D=(-5)^2 - 4\times2\times(-52)=25 + 416 = 441
\]
Then, find \( w \):
\[
w=\frac{5\pm\sqrt{441}}{4}=\frac{5\pm21}{4}
\]
We have two solutions:

• \( w=\frac{5 + 21}{4}=\frac{26}{4}=6.5 \)
• \( w=\frac{5 - 21}{4}=\frac{-16}{4}=-4 \)

Since width cannot be negative, we take \( w = 6.5 \) yards.

Step4: Find the length

Substitute \( w = 6.5 \) into the length formula \( l = 2w - 5 \):
\[
l = 2\times6.5 - 5 = 13 - 5 = 8
\]

Answer:

Length: \( 8 \) yd
Width: \( 6.5 \) yd