A rectangular pool measuring 6 feet by 12 feet is surrounded by a walkway. The width of the walkway is the same on all four sides of the pool. If the total area of the walkway and pool is 520 square feet, what is the number of feet in the width of the walkway?
(6+2w)(12+2w) = 520
w = 7
To solve this problem, let's denote the width of the walkway as 'x' feet.
The length of the pool, including the walkway, can be calculated as follows:
Length of pool = length of pool + 2(width of walkway) = 6 + 2x feet
Similarly, the width of the pool, including the walkway, can be calculated as:
Width of pool = width of pool + 2(width of walkway) = 12 + 2x feet
Now, the area of the pool, including the walkway, can be calculated as:
Area of pool = Length of pool * Width of pool = (6 + 2x)(12 + 2x) square feet
Given that the total area of the pool and walkway is 520 square feet, we can set up the equation:
Area of pool + Area of walkway = 520 square feet
(6 + 2x)(12 + 2x) + (6 * 12) = 520
Expanding and simplifying the equation:
72 + 12x + 12x + 4x^2 + 72 = 520
4x^2 + 24x + 144 = 520
4x^2 + 24x - 376 = 0
To simplify further, divide all terms by 4:
x^2 + 6x - 94 = 0
This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. In this case, we'll use factoring.
(x + 16)(x - 10) = 0
Setting each factor equal to zero:
x + 16 = 0 or x - 10 = 0
x = -16 or x = 10
Since we're dealing with a physical measurement, we discard the negative solution:
x = 10
Therefore, the width of the walkway is 10 feet.
To solve this problem, we can set up an equation based on the given information.
Let's assume the width of the walkway is "x" feet.
The length of the pool, including the walkway, will be the length of the pool plus twice the width of the walkway. So, the total length will be (12 + 2x) feet.
Similarly, the width of the pool, including the walkway, will be the width of the pool plus twice the width of the walkway. So, the total width will be (6 + 2x) feet.
The area of the pool, including the walkway, will be the product of the length and width: (12 + 2x)(6 + 2x).
Now, we can set up the equation based on the given information:
(12 + 2x)(6 + 2x) - 6 * 12 = 520
Simplifying this equation will give us a quadratic equation, which we can solve to find the value of "x," representing the width of the walkway.
To solve the equation, you can follow these steps:
1. Expand the equation: (12 + 2x)(6 + 2x) - 6 * 12 = 520
Simplify this expression: 72 + 24x + 12x + 4x^2 - 72 = 520
Combine like terms: 4x^2 + 36x = 520
2. Move all terms to one side to have the quadratic equation in standard form:
4x^2 + 36x - 520 = 0
3. Factor the quadratic equation:
Start by finding the factors of the constant term (-520) that add up to the coefficient of the "x" term (36):
The factors of 520 are: (1, 520), (2, 260), (4, 130), (5, 104), (8, 65), (10, 52), (13, 40), (20, 26)
The factors that add up to 36 are: (4, 130), (8, 65), (13, 40), (20, 26)
So, the factored form of the quadratic equation is:
(4x - 20)(x + 26) = 0
4. Set each factor equal to zero and solve for "x":
4x - 20 = 0 -> 4x = 20 -> x = 5
x + 26 = 0 -> x = -26 (Discard this solution because we are dealing with the width of the walkway, which cannot be negative.)
Therefore, the width of the walkway is 5 feet.