Question

the area of a rectangle is 27 ft², and the length of the rectangle is 3 ft less than twice the width. find the dimensions of the rectangle. length : □ ft width : □ ft

Explanation:

Step1: Set up variables

Let the width of the rectangle be \( w\) ft. Then the length \( l=(2w - 3)\) ft.

Step2: Use the area formula

The area formula for a rectangle is \( A=l\times w\). Given \( A = 27\) ft², we substitute \( l\) and \( w\) into the formula: \( 27=(2w - 3)\times w\).
Expanding gives \( 27 = 2w^{2}-3w\).
Rearranging to standard quadratic form: \( 2w^{2}-3w - 27=0\).

Step3: Solve the quadratic equation

We can solve \( 2w^{2}-3w - 27 = 0\) using the quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \( a = 2\), \( b=-3\), \( c=-27\).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(-3)^{2}-4\times2\times(-27)=9 + 216=225\).
Then \( w=\frac{3\pm\sqrt{225}}{4}=\frac{3\pm15}{4}\).
We have two solutions for \( w\):
\( w_{1}=\frac{3 + 15}{4}=\frac{18}{4}=4.5\) and \( w_{2}=\frac{3-15}{4}=\frac{-12}{4}=- 3\).
Since width cannot be negative, \( w = 4.5\) ft.

Step4: Find the length

Substitute \( w = 4.5\) into \( l=2w - 3\).
\( l=2\times4.5-3=9 - 3=6\) ft.

Answer:

Length: \(6\) ft, Width: \(4.5\) ft