A 23-ft by 47-ft rectangular swimming pool is surrounded by a walkway of uniform width. If the total area of the walkway is 456ft^2, how wide is the walkway?

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