Question

question 46 details the width of a rectangle is 2 feet longer than four times its length, and its area is 72. if you use l to represent the length of the rectangle, then an expression representing the width is w = the area of the rectangle is 72 square feet. an equation for the area of the rectangle in terms of the length l is: the length of the rectangle is feet and the width is feet.

Explanation:

Step1: Find the expression for width

Given that the width \(W\) is 2 feet longer than four times the length \(L\).
So, \(W = 4L+2\)

Step2: Form the area equation

The area of a rectangle \(A = L\times W\). Since \(A = 72\) and \(W=4L + 2\), we substitute \(W\) into the area formula.
\(72=L\times(4L + 2)\)
Expand the right - hand side: \(72 = 4L^{2}+2L\)
Rearrange to get a quadratic equation: \(4L^{2}+2L-72 = 0\)
Divide through by 2: \(2L^{2}+L - 36=0\)

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c=0\) (here \(a = 2\), \(b = 1\), \(c=-36\)), we use the quadratic formula \(L=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
First, calculate the discriminant \(\Delta=b^{2}-4ac=(1)^{2}-4\times2\times(-36)=1 + 288=289\)
Then \(L=\frac{-1\pm\sqrt{289}}{2\times2}=\frac{-1\pm17}{4}\)
We have two solutions for \(L\):
\(L_{1}=\frac{-1 + 17}{4}=\frac{16}{4}=4\)
\(L_{2}=\frac{-1-17}{4}=\frac{-18}{4}=-4.5\)
Since length cannot be negative, we take \(L = 4\)

Step4: Find the width

Substitute \(L = 4\) into \(W=4L + 2\)
\(W=4\times4+2=16 + 2=18\)

Answer:

• The expression for the width: \(W = 4L+2\)
• The area equation: \(4L^{2}+2L=72\) (or \(2L^{2}+L - 36 = 0\))
• The length of the rectangle is \(4\) feet and the width is \(18\) feet.