Mr. Armstrong wishes to build a deck of uniform width around his pool. The pool is rectangular and has dimensions
6m by 9m. If the area of the deck is the same as the area of the pool, determine the width of the deck.
To determine the width of the deck, we need to find the area of both the pool and the deck.
The area of the pool is given by length multiplied by width: 6m * 9m = 54m^2.
Since the area of the deck is the same as the area of the pool, the area of the deck is also 54m^2.
Let's assume the width of the deck is x.
The overall size of the pool and the deck together is increased by 2 times the width on each side.
So, the total dimensions of the pool and deck together would be (6 + 2x) by (9 + 2x).
The area of the pool and deck together would be (6 + 2x) * (9 + 2x).
Since the area of the pool and deck together is the same as the area of the pool, we can set up the following equation:
(6 + 2x) * (9 + 2x) = 54
Expanding the equation gives us:
54 + 12x + 18x + 4x^2 = 54
Combine like terms:
4x^2 + 30x + 54 = 54
Simplify:
4x^2 + 30x = 0
Factor out 2x:
2x(2x + 15) = 0
Set each factor equal to zero and solve for x:
2x = 0 or 2x + 15 = 0
If 2x = 0, then x = 0, but a deck with zero width is not possible.
So, if 2x + 15 = 0, then 2x = -15, and x = -7.5.
Since the width cannot be negative, we can discard the negative solution.
Therefore, the width of the deck is x = 0m.
To determine the width of the deck, we need to find the difference in area between the pool and the deck.
The pool has dimensions of 6m by 9m, so its area is 6m * 9m = 54m².
Let's assume the width of the deck is x meters.
The length of the pool with the deck will be the original length plus two times the width of the deck. So, the length of the pool with the deck is 9m + 2x.
The width of the pool with the deck will be the original width plus two times the width of the deck. So, the width of the pool with the deck is 6m + 2x.
The area of the pool with the deck is (9m + 2x) * (6m + 2x).
We know that the area of the deck is equal to the area of the pool, so we can set up the following equation:
(9m + 2x) * (6m + 2x) = 54m²
Expanding this equation:
54m² = 54m² + 18mx + 12mx + 4x²
Simplifying the equation:
0 = 30mx + 4x²
Dividing both sides by 2x:
0 = 15m + 2x
Rearranging the equation:
2x = -15m
Dividing both sides by 2:
x = -7.5m
We have a negative value for x, which doesn't make sense in this context. The width cannot be negative.
Therefore, there is no width of the deck that will result in the area of the deck being equal to the area of the pool.