Let's assume the width of the brick patio is "w" meters. According to the given information, the length of the patio is twice the width, so the length is "2w" meters.
The garden is 1.5 meters wide all around the patio. Therefore, the dimensions of the garden will be "w + 3" meters in width (1.5 meters on one side and 1.5 meters on the other side) and "2w + 3" meters in length (1.5 meters on one side and 1.5 meters on the other side).
The area of the garden is given as 54 square meters. We can set up the equation:
Area of the garden = (width of the garden) x (length of the garden)
54 m² = (w + 3) m * (2w + 3) m
Now, we can solve this equation for "w":
54 = (w + 3)(2w + 3)
Expand the equation:
54 = 2w^2 + 3w + 6w + 9
Combine like terms:
2w^2 + 9w + 9 = 54
Subtract 54 from both sides to set the equation to zero:
2w^2 + 9w + 9 - 54 = 0
2w^2 + 9w - 45 = 0
Now, we need to solve this quadratic equation for "w." We can simplify it by dividing the entire equation by 2 to make it easier to solve:
w^2 + 4.5w - 22.5 = 0
Next, we can solve for "w" using the quadratic formula:
w = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 1, b = 4.5, and c = -22.5. Plug these values into the formula:
w = (-4.5 ± √(4.5² - 4 * 1 * (-22.5))) / (2 * 1)
w = (-4.5 ± √(20.25 + 90)) / 2
w = (-4.5 ± √110.25) / 2
w = (-4.5 ± 10.5) / 2
Now, we have two possible solutions for "w":
w = (-4.5 + 10.5) / 2 = 6 / 2 = 3 meters
w = (-4.5 - 10.5) / 2 = -15 / 2 (not a valid solution since width cannot be negative in this context)
So, the width of the brick patio is 3 meters, and the length is 2w = 6 meters. Therefore, the dimensions of the patio are 3 meters in width and 6 meters in length.