### Part (a): Simplification
Let's simplify the problem given in part (b) step by step.
### Part (b): Solving the Problem
#### Step 1: Understanding the Problem
We are given a wire of length 38 cm that is bent into the shape of a rectangle. The length of the rectangle is 7 cm more than the width. We need to find:
1. The width and length of the rectangle.
2. The area of the rectangle.
3. Given farone is 720 metres, find half of the length of the rope.
#### Step 2: Representing the Variables
Let's denote:
- The width of the rectangle by [tex]\( w \)[/tex] cm.
- The length of the rectangle by [tex]\( l \)[/tex] cm.
According to the problem, the length [tex]\( l \)[/tex] is 7 cm more than the width [tex]\( w \)[/tex]:
[tex]\[ l = w + 7 \][/tex]
#### Step 3: Using Perimeter to Formulate Equations
The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[ P = 2 \times (l + w) \][/tex]
Given that the wire length (which is the perimeter) is 38 cm, we can write:
[tex]\[ 2 \times (l + w) = 38 \][/tex]
Substitute [tex]\( l = w + 7 \)[/tex] into the perimeter equation:
[tex]\[ 2 \times ((w + 7) + w) = 38 \][/tex]
Simplify and solve for [tex]\( w \)[/tex]:
[tex]\[ 2 \times (2w + 7) = 38 \][/tex]
[tex]\[ 4w + 14 = 38 \][/tex]
[tex]\[ 4w = 24 \][/tex]
[tex]\[ w = 6 \][/tex]
Thus, the width of the rectangle is [tex]\( w = 6 \)[/tex] cm.
#### Step 4: Calculate the Length
Using [tex]\( w = 6 \)[/tex] cm, we find the length [tex]\( l \)[/tex]:
[tex]\[ l = w + 7 \][/tex]
[tex]\[ l = 6 + 7 \][/tex]
[tex]\[ l = 13 \][/tex] cm
#### Step 5: Calculate the Area
The area [tex]\( A \)[/tex] of the rectangle is given by:
[tex]\[ A = l \times w \][/tex]
[tex]\[ A = 13 \times 6 \][/tex]
[tex]\[ A = 78 \][/tex] square cm
#### Step 6: Find Half of the Length of the Rope
Given farone is 720 metres, we need to find half of the length of the rope:
[tex]\[ \text{Half of 720 metres} = \frac{720}{2} = 360 \][/tex] metres
### Summary
1. The width of the rectangle is [tex]\( 6 \)[/tex] cm.
2. The length of the rectangle is [tex]\( 13 \)[/tex] cm.
3. The area of the rectangle is [tex]\( 78 \)[/tex] square cm.
4. Half of the length of the rope (farone) is [tex]\( 360 \)[/tex] metres.
