Certainly! Let's break this problem down step-by-step.
1. Dimensions of the Garden:
- Let's denote the width of the garden as [tex]\( x \)[/tex].
- The length of the garden is three times the width, so the length is [tex]\( 3x \)[/tex].
2. Dimensions of the Entire Area Including the Walkway:
- The walkway surrounds the garden and is 4 feet wide on each side.
- Therefore, the total width of the garden and walkway combined is [tex]\( x + 2 \times 4 = x + 8 \)[/tex] feet.
- Similarly, the total length of the garden and walkway combined is [tex]\( 3x + 2 \times 4 = 3x + 8 \)[/tex] feet.
3. Area Calculations:
- Area of the Garden:
- The area of the garden alone is [tex]\( \text{length} \times \text{width} = 3x \times x = 3x^2 \)[/tex].
- Area of the Entire Area (Garden + Walkway):
- The area of the entire space, including the garden and the walkway, is given by the product of the total length and width:
[tex]\[
\text{Area}_{\text{total}} = (3x + 8) \times (x + 8)
\][/tex]
Let's expand this product:
[tex]\[
\text{Area}_{\text{total}} = (3x + 8)(x + 8)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
\text{Area}_{\text{total}} = 3x \cdot x + 3x \cdot 8 + 8 \cdot x + 8 \cdot 8
\][/tex]
[tex]\[
\text{Area}_{\text{total}} = 3x^2 + 24x + 8x + 64
\][/tex]
Combine like terms:
[tex]\[
\text{Area}_{\text{total}} = 3x^2 + 32x + 64
\][/tex]
- Area of the Walkway Alone:
- The area of the walkway alone is the difference between the total area and the area of the garden:
[tex]\[
\text{Area}_{\text{walkway}} = \text{Area}_{\text{total}} - \text{Area}_{\text{garden}}
\][/tex]
- Substitute the expressions for the areas:
[tex]\[
\text{Area}_{\text{walkway}} = (3x^2 + 32x + 64) - 3x^2
\][/tex]
- Simplify by subtracting the common terms:
[tex]\[
\text{Area}_{\text{walkway}} = 3x^2 + 32x + 64 - 3x^2
\][/tex]
[tex]\[
\text{Area}_{\text{walkway}} = 32x + 64
\][/tex]
So, the polynomial expressions are:
- Area of the garden: [tex]\( 3x^2 \)[/tex]
- Area of the entire space (garden + walkway): [tex]\( 3x^2 + 32x + 64 \)[/tex]
- Area of the walkway alone: [tex]\( 32x + 64 \)[/tex]
