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 = \boxed{} centimeters
width = \boxed{} centimeters

Explanation:

Step1: Define variables

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

Step2: Use area formula

The area of a rectangle is \( A = l\times w \). We know \( A = 330 \), so substitute \( w = l - 7 \) into the formula:
\( l(l - 7)=330 \)
Expand to get \( l^{2}-7l - 330 = 0 \)

Step3: Solve quadratic equation

Factor the quadratic equation: \( l^{2}-7l - 330=(l - 22)(l + 15)=0 \)
Set each factor to zero: \( l - 22 = 0 \) or \( l + 15 = 0 \)
Solutions are \( l = 22 \) or \( l=-15 \). Since length can't be negative, \( l = 22 \)

Step4: Find width

Width \( w = l - 7 = 22 - 7 = 15 \)

Answer:

length = 22 centimeters
width = 15 centimeters