Sure, let's solve the problem step-by-step:
### Step 1: Define the Dimensions of the Pool
The pool has a length of 50 meters and a width of 25 meters. The area of the pool can be calculated as:
[tex]\[ \text{Pool Area} = \text{Length} \times \text{Width} = 50 \, \text{m} \times 25 \, \text{m} = 1250 \, \text{m}^2 \][/tex]
### Step 2: Determine the Dimensions of the Outer Rectangle
The walkway surrounds the pool on all sides, and the width of the walkway is [tex]\( x \)[/tex] meters. Therefore:
- The total length of the outer rectangle (which includes the pool and the walkway) is [tex]\( 50 + 2x \)[/tex] meters.
- The total width of the outer rectangle is [tex]\( 25 + 2x \)[/tex] meters.
### Step 3: Calculate the Area of the Outer Rectangle
The area of the outer rectangle can be determined using its length and width:
[tex]\[ \text{Outer Area} = (\text{Total Length}) \times (\text{Total Width}) = (50 + 2x) \times (25 + 2x) \][/tex]
### Step 4: Determine the Area of the Walkway
The area of the walkway is the difference between the area of the outer rectangle and the area of the pool. This can be calculated as follows:
[tex]\[ \text{Walkway Area} = \text{Outer Area} - \text{Pool Area} = (50 + 2x)(25 + 2x) - 1250 \][/tex]
### Summary
- The area of the pool is [tex]\( 1250 \, \text{m}^2 \)[/tex].
- The area of the outer rectangle is [tex]\( (50 + 2x)(25 + 2x) \)[/tex].
- The area of the walkway is [tex]\( (50 + 2x)(25 + 2x) - 1250 \)[/tex].
So, the final expressions are:
[tex]\[ \text{Walkway Area} = (50 + 2x)(25 + 2x) - 1250 \][/tex]
[tex]\[ \text{Outer Area} = (50 + 2x)(25 + 2x) \][/tex]
[tex]\[ \text{Pool Area} = 1250 \, \text{m}^2 \][/tex]
Therefore, the area of the cement walkway around a 50 m by 25 m pool with a walkway width of [tex]\( x \)[/tex] meters is given by:
[tex]\[ (2x + 25) \times (2x + 50) - 1250 \, \text{m}^2 \][/tex]
