Question

the length of a rectangle is 4 more than the width. the area of the rectangle is 10 square yards. what is the length of the rectangle? let (w =) width and (4 + w=) length. which equation represents the situation? using the zero - product property, the equation is what is the length of the rectangle? yards

Explanation:

Step1: Set up area - formula equation

The area formula of a rectangle is $A = length\times width$. Given $width = w$, $length=w + 4$ and $A = 16$. So the equation is $w(w + 4)=16$. Expand it to get $w^{2}+4w-16 = 0$.

Step2: Solve the quadratic equation

Using the quadratic formula $w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ for the quadratic equation $aw^{2}+bw + c = 0$. Here, $a = 1$, $b = 4$, $c=-16$. First, calculate the discriminant $\Delta=b^{2}-4ac=(4)^{2}-4\times1\times(-16)=16 + 64 = 80$. Then $w=\frac{-4\pm\sqrt{80}}{2}=\frac{-4\pm4\sqrt{5}}{2}=-2\pm2\sqrt{5}$. Since the width cannot be negative, $w=-2 + 2\sqrt{5}$.

Step3: Find the length

The length $l=w + 4$. Substitute $w=-2 + 2\sqrt{5}$ into it, $l=-2+2\sqrt{5}+4=2 + 2\sqrt{5}$.

Answer:

$2 + 2\sqrt{5}$