A rectangular pool is 20 feet wide and 50 feet long. A deck used for sunning surrounds the pool. The deck is the same width all the way around the pool. The total area of the deck is 456 square feet. How wide is the deck around the pool? Show your work
Let x be the width of the deck around the pool.
The total area of the deck can be represented as the area of the larger rectangle (with dimensions 20 + 2x and 50 + 2x) minus the area of the pool itself (20*50).
So, the area of the deck can be expressed as:
(20 + 2x)(50 + 2x) - 20*50 = 456
Expanding and simplifying:
1000 + 40x + 100x + 4x^2 - 1000 = 456
4x^2 + 140x - 456 = 0
Divide the entire equation by 4 to simplify:
x^2 + 35x - 114 = 0
Now, we need to solve for x using the quadratic formula:
x = (-B ± sqrt(B^2 - 4AC)) / 2A
where A = 1, B = 35, C = -114
x = (-35 ± sqrt(35^2 - 4*1*(-114))) / 2*1
x = (-35 ± sqrt(1225 + 456)) / 2
x = (-35 ± sqrt(1681)) / 2
x = (-35 ± 41) / 2
x = (6 or -76)
Since the width can't be negative, we use x = 6 feet.
Therefore, the width of the deck around the pool is 6 feet.