A rectangular pool measures 12 feet by 20 feet. A patio of uniform width is built around the pool. The area of both the pool and the patio together is 560 square feet. Find the width of the patio.
dimensions of pool including the patio is (12+2x) by (20 + 2x), where x is the width of the patio around the pool
(12+2x)(20+2x) + 12(20) = 560
240 + 24x + 40x + 4x^2 + 240 - 560 = 0
4x^2 + 64x - 80 = 0
x^2 + 16x - 20 = 0
solve using the quadratic formula since it doesn't factor.
To find the width of the patio, we need to subtract the area of the pool from the total area of the pool and the patio.
1. Start with the total area of the pool and patio together. We are given that it is 560 square feet.
Total area = 560 square feet
2. Next, subtract the area of the pool from the total area to find the area of the patio.
Area of patio = Total area - Area of pool
3. Calculate the area of the pool by multiplying its length and width.
Area of pool = Length of pool × Width of pool
= 12 feet × 20 feet
4. Substitute the known values into the equation to find the area of the pool.
Area of pool = 240 square feet
5. Plug the area of the pool back into the equation to find the area of the patio.
Area of patio = 560 square feet - 240 square feet
= 320 square feet
6. We know that the patio has a uniform width, so we can assume that it is the same width all around. Let's call this width "W".
7. Calculate the width of the patio by dividing the area of the patio by its length.
Width of patio = Area of patio / Overall length
= 320 square feet / 20 feet
= 16 feet
Therefore, the width of the patio is 16 feet.
To find the width of the patio, we need to set up an equation using the given information.
Let's assume the width of the patio is "x" feet.
The length of the pool including the width of the patio will be 12 + 2x, and the width of the pool including the width of the patio will be 20 + 2x.
The area of both the pool and the patio together is given as 560 square feet:
(12 + 2x) * (20 + 2x) = 560
Now we can solve the equation to find the width of the patio.
First, let's simplify the equation:
(12 + 2x) * (20 + 2x) = 560
240 + 24x + 40x + 4x^2 = 560
4x^2 + 64x + 240 - 560 = 0
4x^2 + 64x - 320 = 0
Divide the equation by 4 to simplify it further:
x^2 + 16x - 80 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use factoring:
(x + 20)(x - 4) = 0
So either (x + 20) = 0 or (x - 4) = 0
If x + 20 = 0, then x = -20, which doesn't make sense in the context of the problem since we cannot have a negative width.
If x - 4 = 0, then x = 4, which is the width of the patio.
Therefore, the width of the patio is 4 feet.