A 6 m by 9 m swimming pool is surrounded by a deck which has a constant width. How wide is the deck if its area equals the area of the pool?
I got 6*9=54 so for the deck xy=54 but I don't know where to go.
pool area = 54
whole thing
L = 9+2s
w = 6+2s
area of whole thing = (9+2s)(6+2s)
so
(9+2s)(6+2s) = 2 * 54 = 108
54 + 30 s + 4 s^2 - 108 = 0
4 s^2 + 30 s - 54 = 0
2 s^2 + 15 s - 27 = 0
(2s-3)(s+9) = 0
s = 3/2 = 1.5
To solve this problem, let's approach it step by step:
1. Start by calculating the total area of the pool:
The length of the pool is 6m and the width is 9m, so the area of the pool is 6m * 9m = 54m².
2. Now, let's assume the width of the deck is 'x' meters. This means that the overall length of the pool, including the deck, would be 6m + 2x (since the deck surrounds the pool on both sides), and the overall width of the pool, including the deck, would be 9m + 2x.
3. To find the total area of the pool and the deck combined, we can use the formula: Total area = Length * Width.
Therefore, the total area of the pool and the deck combined would be (6m + 2x)(9m + 2x).
4. According to the problem statement, the area of the deck should be equal to the area of the pool. So, we can set up an equation: (6m + 2x)(9m + 2x) = 54m².
5. Expand the equation: (6m * 9m) + (6m * 2x) + (2x * 9m) + (2x * 2x) = 54m².
6. Simplify the equation further: 54m² + 12mx + 18mx + 4x² = 54m².
7. Cancel out the common term '54m²' from both sides of the equation: 12mx + 18mx + 4x² = 0.
8. Combine like terms: 30mx + 4x² = 0.
9. Set the equation equal to zero to solve for 'x': 4x² + 30mx = 0.
10. Factor out 'x' to solve for 'x': x(4x + 30m) = 0.
11. Since we are looking for the width of the deck, 'x' cannot be zero. Therefore, we only consider the second term (4x + 30m) in the parentheses: 4x + 30m = 0.
12. Solve for 'x': 4x = -30m.
x = -30m / 4.
13. Simplify the equation: x = -7.5m.
Regarding the solution, we find that the width of the deck is -7.5 meters. However, a negative value for width doesn't make sense in this context. Therefore, we discard the negative solution.
Hence, the width of the deck is 7.5 meters.