Question

the length of a rectangle is 1 less than twice the width. the area of the rectangle is 28 square feet. which equation represents the situation? what is the length of the rectangle?

Explanation:

Step1: Define variables

Let the width of the rectangle be $w$. Then the length $l = 2w - 1$.

Step2: Use area formula

The area formula of a rectangle is $A=l\times w$. Given $A = 28$, we substitute $l$ and $A$ into the formula: $28=(2w - 1)\times w$. Expanding gives $28 = 2w^{2}-w$, or $2w^{2}-w - 28=0$.

Step3: Solve quadratic equation for width

For the quadratic equation $2w^{2}-w - 28 = 0$, where $a = 2$, $b=-1$, $c=-28$. Using the quadratic formula $w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{1\pm\sqrt{(-1)^{2}-4\times2\times(-28)}}{2\times2}=\frac{1\pm\sqrt{1 + 224}}{4}=\frac{1\pm\sqrt{225}}{4}=\frac{1\pm15}{4}$. We get two solutions for $w$: $w=\frac{1 + 15}{4}=4$ and $w=\frac{1-15}{4}=-\frac{7}{2}$. Since width cannot be negative, $w = 4$.

Step4: Find the length

Substitute $w = 4$ into the length formula $l=2w - 1$. Then $l=2\times4-1=7$.

Answer:

The equation representing the situation is $2w^{2}-w - 28 = 0$. The length of the rectangle is 7 feet.