Question

a rectangle has an area of 126 square feet. the length is 3 more than 3 times the width. find the length and width, in feet, of the rectangle.
show your work here

Explanation:

Step1: Define variables

Let the width be \( w \). Then the length \( l=3w + 3 \).

Step2: Use area formula

The area of a rectangle \( A=l\times w \). Substitute \( l = 3w+3 \) and \( A = 126 \) into the formula: \( 126=(3w + 3)\times w \).
Expand: \( 126=3w^{2}+3w \).
Divide both sides by 3: \( 42=w^{2}+w \).
Rearrange to standard quadratic form: \( w^{2}+w - 42=0 \).

Step3: Solve quadratic equation

Factor the quadratic: \( w^{2}+w - 42=(w + 7)(w- 6)=0 \).
Set each factor equal to zero: \( w+7 = 0\) gives \( w=-7 \) (rejected as width cannot be negative), \( w - 6=0\) gives \( w = 6 \).

Step4: Find the length

Substitute \( w = 6 \) into \( l=3w + 3 \). Then \( l=3\times6+3=18 + 3=21 \).

Answer:

The width of the rectangle is \( 6 \) feet and the length is \( 21 \) feet.