Final answer:
To find the width of the deck around the pool, calculate the total area and subtract the area of the pool. Then, use the quadratic equation to solve for the width of the deck, resulting in a width of 4 feet.
Step-by-step explanation:
To solve for the width of the deck surrounding the pool, we start with the dimensions of the pool which are 18 feet by 24 feet. This gives us an area of the pool itself which is 18 ft × 24 ft = 432 square feet. Since the total area of the deck only is 400 square feet, the total area including both the pool and the deck must be 432 square feet + 400 square feet = 832 square feet.
Let w represent the width of the deck. The dimensions of the pool including the deck become (18 ft + 2w) × (24 ft + 2w). We set up the equation (18 ft + 2w)(24 ft + 2w) = 832 square feet and solve for w. Distributing and simplifying the equation, we get:
18 ft × 24 ft + 2w(18 ft) + 2w(24 ft) + 4w^2 = 832 square feet
432 square feet + 36w + 48w + 4w^2 = 832 square feet
4w^2 + 84w + 432 = 832
4w^2 + 84w - 400 = 0
Dividing all terms by 4 we get:
w^2 + 21w - 100 = 0
Factoring the quadratic equation, we find:
(w + 25)(w - 4) = 0
Thus, w = -25 or w = 4. Since the width cannot be negative, the width of the deck is 4 feet.
Answer: B. 4 feet