Question

a rectangle has a width that is 7 centimeters less than its length, and its area is 330 square centimeters. what are the dimensions of the rectangle? length = □ centimeters width = □ centimeters

Explanation:

Step1: Define variables

Let the length of the rectangle be \( x \) centimeters. Then the width is \( x - 7 \) centimeters.

Step2: Use area formula

The area of a rectangle is \( \text{length} \times \text{width} \), so we have the equation \( x(x - 7)=330 \).

Step3: Expand and rearrange

Expanding gives \( x^{2}-7x = 330 \), and rearranging to standard quadratic form: \( x^{2}-7x - 330=0 \).

Step4: Factor the quadratic

We need two numbers that multiply to - 330 and add to - 7. The numbers are - 22 and 15. So, \( (x - 22)(x + 15)=0 \).

Step5: Solve for x

Setting each factor equal to zero: \( x - 22 = 0 \) or \( x+15 = 0 \). Solving these, we get \( x = 22 \) or \( x=-15 \). Since length can't be negative, we take \( x = 22 \).

Step6: Find width

The width is \( x - 7=22 - 7 = 15 \) centimeters.

Answer:

length = 22 centimeters
width = 15 centimeters