Question

the length of a rectangle is 6 inches longer than it is wide. if the area is 72 square inches, what are the dimensions of the rectangle? the width is 6 inches. the length is 12 inches.

Explanation:

Step1: Define variables

Let the width of the rectangle be $x$ inches. Then the length is $x + 6$ inches.

Step2: Set up area - formula equation

The area formula for a rectangle is $A=length\times width$. Given $A = 72$ square inches, we have the equation $x(x + 6)=72$.

Step3: Expand and rearrange the equation

Expand $x(x + 6)$ to get $x^{2}+6x=72$. Rearrange it to the standard quadratic - form $x^{2}+6x - 72=0$.

Step4: Solve the quadratic equation

We can factor the quadratic equation $x^{2}+6x - 72=(x + 12)(x - 6)=0$.
Setting each factor equal to zero gives $x+12 = 0$ or $x - 6=0$.
Solving $x+12 = 0$ gives $x=-12$, and solving $x - 6=0$ gives $x = 6$. Since the width cannot be negative, we take $x = 6$.

Step5: Find the length

The length is $x + 6$. Substituting $x = 6$ into $x + 6$, we get $6+6=12$ inches.

Answer:

The width is 6 inches.
The length is 12 inches.