To solve the problem of finding the area of a rectangle formed by a wire of length 38 cm, where the length of the rectangle is 7 cm more than the width, let's follow these steps:
1. Define the variables:
- Let the width of the rectangle be [tex]\( w \)[/tex] cm.
- Let the length of the rectangle be [tex]\( l \)[/tex] cm.
- Given that the length is 7 cm more than the width, we can express the length as [tex]\( l = w + 7 \)[/tex].
2. Write the formula for the perimeter of the rectangle:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by the formula:
[tex]\[
P = 2 \times \text{(length + width)}
\][/tex]
3. Substitute the given perimeter and the relationship between length and width into the formula:
- We know the perimeter is 38 cm, so:
[tex]\[
38 = 2 \times (w + (w + 7))
\][/tex]
4. Simplify the equation:
- Combine like terms inside the parentheses:
[tex]\[
38 = 2 \times (2w + 7)
\][/tex]
- Distribute the 2:
[tex]\[
38 = 4w + 14
\][/tex]
5. Solve for [tex]\( w \)[/tex]:
- Subtract 14 from both sides to isolate the term with [tex]\( w \)[/tex]:
[tex]\[
38 - 14 = 4w
\][/tex]
- Simplify this:
[tex]\[
24 = 4w
\][/tex]
- Divide both sides by 4:
[tex]\[
w = 6
\][/tex]
6. Find the length [tex]\( l \)[/tex]:
- Recall that the length [tex]\( l \)[/tex] is 7 cm more than the width:
[tex]\[
l = w + 7 = 6 + 7 = 13
\][/tex]
7. Calculate the area of the rectangle:
- The area [tex]\( A \)[/tex] of a rectangle is given by the formula:
[tex]\[
A = \text{length} \times \text{width}
\][/tex]
- Substitute the values of the length and width:
[tex]\[
A = 13 \times 6 = 78
\][/tex]
Thus, the area of the rectangle is [tex]\( 78 \)[/tex] square centimeters.
