To describe the situation algebraically and calculate the area of the pathway, follow these steps:

Step 1: Define the variables.

Let:

• $w$ represent the width of the pool.
• $l$ represent the length of the pool.

Since the length is 4 times the width, we have the relationship: $l = 4w$

Step 2: Set up the equation for the perimeter of the pool.

The perimeter of the pool is given as 120 meters. The perimeter of a rectangle is: $P = 2l + 2w$ Substituting $l = 4w$ into this equation: $120 = 2(4w) + 2w$ $120 = 8w + 2w$ $120 = 10w$ Solve for $w$: $w = \frac{120}{10} = 12 \text{ meters}$ Now substitute $w = 12$ into $l = 4w$: $l = 4(12) = 48 \text{ meters}$

Step 3: Determine the new perimeter including the pathway.

The pathway has a width of 2 meters around the pool, so the new dimensions of the pool with the pathway are:

• New width = $w + 2(2) = 12 + 4 = 16$ meters
• New length = $l + 2(2) = 48 + 4 = 52$ meters

Now, the new perimeter is: $P_{\text{new}} = 2(52) + 2(16) = 104 + 32 = 136 \text{ meters}$

Step 4: Calculate the area of the pathway.

The total area of the pool with the pathway is: $\text{Area}_{\text{total}} = 52 \times 16 = 832 \text{ square meters}$ The area of the pool alone is: $\text{Area}_{\text{pool}} = 48 \times 12 = 576 \text{ square meters}$ Thus, the area of the pathway is: $\text{Area}_{\text{pathway}} = \text{Area}_{\text{total}} - \text{Area}_{\text{pool}} = 832 - 576 = 256 \text{ square meters}$

Final Answer:

• Algebraic equation for new perimeter: $P_{\text{new}} = 136 \text{ meters}$
• Area of the pathway: 256 square meters.

Do you want more details or have any questions?

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Here are 5 related questions:

1. What would the new perimeter be if the pathway was 3 meters wide?
2. How does the area of the pool change if the width is doubled?
3. How much additional material is needed to extend the pathway width by 1 meter?
4. What is the total cost of covering the pathway if the cost per square meter is $50?
5. How much water can the pool hold if the depth is 2 meters?

Tip: When working with perimeter and area problems, always make sure to clearly define your variables and relationships!