Answer:
The wide of the walkway is 3 feet
Step-by-step explanation:
* Lets explain how to solve the problem
- A 23-ft by 47-ft rectangular swimming pool is surrounded by a
walkway of uniform width
- Assume that the uniform width is x
- That means each dimensions of the the rectangular pool will exceed
by 2x (x from each side)
- The dimensions of the pool with the walkway are 23 + 2x and 47 + 2x
- The total area of the walkway is the difference between the area of
the pool with walkway and the area of the pool
- Area of any rectangle = l × w, where l, w are its dimensions
∴ The area of the pool = 23 × 47 = 1081 ft²
∴ The area of the pool with walkway = (23 + 2x)(47 + 2x)
∴ The area of walkway = (23 + 2x)(47 + 2x) - 1081 ⇒ (1)
∵ The area of the walkway is 456 ft² ⇒ (2)
- Equate (1) and (2)
∴ (23 + 2x)(47 + 2x) - 1081 = 456
- Add 1081 for both sides
∴ (23 + 2x)(47 + 2x) = 1537
- Multiply 2 brackets
∴ (23 × 47) + (23 × 2x) + (2x × 47) + (2x × 2x) = 1537
∴ 1081 + 46x + 94x + 4x² = 1537
∴ 1081 + 140x + 4x² = 1537
- Subtract 1081 from both sides
∴ 4x² + 140x - 456 = 0
- Divide all terms by 4 because 4 is a common factor in all terms
∴ x² + 35x - 114 = 0
- Factorize it
∴ (x - 3)(x + 38) = 0
- Equate each bracket by 0
∴ x - 3 = 0
- Add 3 for both sides
∴ x = 3
OR
∴ x + 38 = 0
- Subtract 38 from both sides
∴ x = -38 ⇒ rejected because no negative answer for dimensions
∴ The value of x is 3
* The wide of the walkway is 3 feet
