Question

the length of a rectangle is 7 yd less than three times the width, and the area of the rectangle is 66 yd². find the dimensions of the rectangle. length: □ yd width: □ yd

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) yd. Then the length \( l=(3w - 7) \) yd.

Step2: Use the area formula

The area of a rectangle is \( A=l\times w \). Substitute \( l = 3w-7 \) and \( A = 66 \) into the formula:
\( (3w - 7)\times w=66 \)
Expand to get \( 3w^{2}-7w - 66 = 0 \)

Step3: Solve the quadratic equation

For a quadratic equation \( ax^{2}+bx + c = 0 \) (\( a = 3 \), \( b=-7 \), \( c = - 66 \)), use the quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \)
First, calculate the discriminant \( \Delta=b^{2}-4ac=(-7)^{2}-4\times3\times(-66)=49 + 792=841 \)
Then \( w=\frac{7\pm\sqrt{841}}{6}=\frac{7\pm29}{6} \)
We have two solutions for \( w \):
\( w_{1}=\frac{7 + 29}{6}=\frac{36}{6}=6 \)
\( w_{2}=\frac{7-29}{6}=\frac{-22}{6}=-\frac{11}{3} \)
Since width cannot be negative, \( w = 6 \) yd.

Step4: Find the length

Substitute \( w = 6 \) into \( l=3w - 7 \), we get \( l=3\times6-7=18 - 7 = 11 \) yd.

Answer:

Length: \( 11 \) yd
Width: \( 6 \) yd