Certainly! Let's go through the problem step-by-step:
1. Identify the Variables:
- Let the width of the rectangle be [tex]\( w \)[/tex] centimeters.
2. Express the Length in Terms of the Width:
- According to the problem, the length of the rectangle is 12 centimeters longer than its width. Hence, if the width is [tex]\( w \)[/tex], then the length [tex]\( l \)[/tex] can be expressed as:
[tex]\[
l = w + 12
\][/tex]
3. Write the Formula for the Area of the Rectangle:
- The area [tex]\( A \)[/tex] of a rectangle is given by the product of its length and width. Thus:
[tex]\[
A = \text{length} \times \text{width}
\][/tex]
4. Substitute the Expressions for Length and Width:
- We have the length [tex]\( l = w + 12 \)[/tex] and the width [tex]\( w \)[/tex]. Substituting these into the area formula, we get:
[tex]\[
A(w) = (w + 12) \times w
\][/tex]
5. Simplify the Expression:
- To make it more formal, we can rewrite it as:
[tex]\[
A(w) = w \times (w + 12)
\][/tex]
Thus, the equation for [tex]\( A(w) \)[/tex], the area of the rectangle in terms of its width [tex]\( w \)[/tex], is:
[tex]\[
A(w) = w \times (w + 12)
\][/tex]
