a 13 ft by 27 ft rectangular swimming pool is surrounded by a walkway of uniform width. if the total area of the walkway is 384 ft^2 how wide is the walkway

To find the width of the walkway, let's denote it as "w". We will first calculate the new dimensions of the overall rectangle (including the walkway).

Length of the new rectangle: 13 ft (original length) + 2w (walkway on both sides)
Width of the new rectangle: 27 ft (original width) + 2w (walkway on both ends)

The area of the new rectangle is equal to the sum of the area of the pool and the area of the walkway.

Total area = (length) x (width)
384 ft^2 = (13 + 2w) ft x (27 + 2w) ft

Expanding the equation:

384 ft^2 = (13 ft)(27 ft) + (13 ft)(2w) + (2w)(27 ft) + (2w)(2w)
384 ft^2 = 351 ft^2 + 26w ft + 54w ft + 4w^2
384 ft^2 = 4w^2 + 80w + 351 ft^2

Now, let's simplify the equation:

4w^2 + 80w - 33 = 0

This equation is a quadratic equation. We can use the quadratic formula to solve for "w".

The quadratic formula is given by:
w = [-b ± √(b^2 - 4ac)] / (2a)

For our equation, a = 4, b = 80, and c = -33.

Substituting the values into the quadratic formula:

w = [-80 ± √(80^2 - 4(4)(-33))] / (2(4))
w = [-80 ± √(6400 + 528)] / 8
w = [-80 ± √(6928)] / 8
w ≈ [-80 ± 83.25] / 8

Simplifying this further, we get two possible solutions:

w ≈ (-80 + 83.25) / 8 ≈ 0.40625 ft (or 4 7/16 inches)
w ≈ (-80 - 83.25) / 8 ≈ -20.40625 ft

Since the width cannot be negative, we disregard the negative solution. Therefore, the approximate width of the walkway is 0.40625 ft (or 4 7/16 inches).

To find the width of the walkway, we need to subtract the area of the inner rectangle (pool) from the area of the outer rectangle (pool + walkway).

First, let's calculate the area of the outer rectangle:

Outer rectangle length = 13 ft + 2 * walkway
Outer rectangle width = 27 ft + 2 * walkway

So the area of the outer rectangle is (13 + 2 * walkway) * (27 + 2 * walkway).

Next, let's calculate the area of the inner rectangle (pool):

Inner rectangle length = 13 ft
Inner rectangle width = 27 ft

So the area of the inner rectangle is 13 ft * 27 ft.

Now, we can set up an equation by subtracting the area of the inner rectangle from the area of the outer rectangle:

(13 + 2 * walkway) * (27 + 2 * walkway) - (13 * 27) = 384 ft^2

Simplify the equation:

(13 + 2 * walkway) * (27 + 2 * walkway) - 351 = 384 ft^2

Expand and rearrange the equation:

(13 * 27) + (2 * 13 * walkway) + (2 * 27 * walkway) + (4 * walkway^2) - 351 = 384 ft^2

Rearrange the equation one more time:

4 * walkway^2 + 26 * walkway + 240 = 384 ft^2

Now, let's solve this quadratic equation for the width of the walkway. We can do this by factoring, completing the square, or using the quadratic formula.

Once you find the values for the walkway width, remember to check if they make sense (i.e., positive values) and round it off to an appropriate decimal point.

The perimeter of the pool is 26 + 54 = 80 feet.

384/80 = 4.8 feet