a rectangle has a width that is 7 centimeters less than its length, and its area is 330 square centimeters…

Explanation:

Step1: Let the length be $x$ cm.

The width is $x - 7$ cm. The area formula for a rectangle is $A=\text{length}\times\text{width}$. So we have the equation $x(x - 7)=330$.

Step2: Expand the left - hand side.

$x^{2}-7x = 330$. Rearrange it to the standard quadratic form $x^{2}-7x-330 = 0$.

Step3: Factor the quadratic equation.

We need to find two numbers that multiply to $- 330$ and add up to $-7$. The numbers are $-22$ and $15$. So $x^{2}-7x - 330=(x - 22)(x + 15)=0$.

Step4: Solve for $x$.

Setting each factor equal to zero gives $x-22 = 0$ or $x + 15=0$. So $x=22$ or $x=-15$. Since length cannot be negative, we take $x = 22$.

Step5: Find the width.

The width is $x-7=22 - 7=15$ cm.

Answer:

Length = 22 centimeters Width = 15 centimeters