Sure! Let's analyze the problem step-by-step.
### 1) Representing the Length
If the width of the vacant lot is represented by [tex]\( w \)[/tex] and the length is 4 meters longer than the width, then the length can be represented as:
[tex]\[ \text{Length} = w + 4 \][/tex]
### 2) Expression for the Area
The area of a rectangle is the product of its length and width. Given the expressions for the width ([tex]\( w \)[/tex]) and length ([tex]\( w + 4 \)[/tex]), the area [tex]\( A \)[/tex] can be written as:
[tex]\[ A = \text{length} \times \text{width} = w \times (w + 4) \][/tex]
### 3) Quadratic Equation for the Area
We know the area of the vacant lot is 12 square meters. Therefore, we can set up the equation:
[tex]\[ w \times (w + 4) = 12 \][/tex]
Expanding and rearranging this equation into the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex], we get:
[tex]\[ w^2 + 4w - 12 = 0 \][/tex]
This is the quadratic equation representing the situation.
### 4) Perimeter Equation for Mang Toti's Fencing Plan
Mang Toti has 16 meters of fencing material and plans to place it all around the rectangular land. The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[ P = 2 \times \text{length} + 2 \times \text{width} \][/tex]
Substituting the expressions for length and width into this equation, we have:
[tex]\[ 2 \times (w + 4) + 2 \times w = 16 \][/tex]
Simplifying the equation step-by-step:
[tex]\[ 2w + 8 + 2w = 16 \][/tex]
[tex]\[ 4w + 8 = 16 \][/tex]
[tex]\[ 4w = 8 \][/tex]
[tex]\[ w = 2 \][/tex]
So, the width [tex]\( w \)[/tex] is 2 meters. Using the width to find the length:
[tex]\[ \text{Length} = w + 4 = 2 + 4 = 6 \][/tex]
Thus, the length is 6 meters.
Summary:
1. The length is represented by [tex]\( w + 4 \)[/tex].
2. The expression for the area is [tex]\( w \times (w + 4) \)[/tex].
3. The quadratic equation is [tex]\( w^2 + 4w - 12 = 0 \)[/tex].
4. The perimeter equation describes the situation as [tex]\( 2 \times (w + 4) + 2 \times w = 16 \)[/tex], leading to width [tex]\( w = 2 \)[/tex] meters and length [tex]\( 6 \)[/tex] meters.
