Question

the length of a rectangle is 7 inches longer than it is wide. if the area is 144 square inches, what are the dimensions of the rectangle? the width, or shorter side is inches the length, or longer side is inches question help: message instructor submit question

Explanation:

Step1: Let the width be $x$ inches.

The length is $x + 7$ inches. The area formula for a rectangle is $A=\text{length}\times\text{width}$. So we have the equation $x(x + 7)=144$.

Step2: Expand the equation.

$x^{2}+7x=144$, which can be rewritten as $x^{2}+7x - 144=0$.

Step3: Solve the quadratic equation.

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 1$, $b=7$, $c=-144$), we use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(7)^{2}-4\times1\times(-144)=49 + 576=625$. Then $x=\frac{-7\pm\sqrt{625}}{2}=\frac{-7\pm25}{2}$. We get two solutions: $x_1=\frac{-7 + 25}{2}=\frac{18}{2}=9$ and $x_2=\frac{-7 - 25}{2}=\frac{-32}{2}=-16$. Since the width cannot be negative, we take $x = 9$.

Step4: Find the length.

The length is $x+7=9 + 7=16$ inches.

Answer:

The width, or shorter side is 9 inches
The length, or longer side is 16 inches