The path around a square flower bed is 3 feet wide. If the area of the flower bed is equal to the area of the path, find the dimensions of the flower bed.
area of bed = x^2
area of whole figure = (x+6)^2
area of path = (x+6)^2 - x^2
so
x^2 + 12 x + 36 -x^2 = x^2
-x^2 + 12 x + 36 = 0
x^2 - 12 x - 36 = 0
(x-6)(x-6) = 0
x = 6
To find the dimensions of the flower bed, we need to set up an equation based on the given information.
Let's assume that the length of one side of the flower bed is "x" feet.
Since the path around the flower bed is 3 feet wide, the length of each side of the flower bed, including the path, would be "x + 6" feet (3 feet on each side).
The area of a square is found by multiplying its length by its width. Therefore, the area of the flower bed (without the path) would be x * x = x^2 square feet.
Since the area of the flower bed is equal to the area of the path, we can set up the equation as follows:
x^2 = (x + 6)^2
Expanding the right side of the equation:
x^2 = x^2 + 12x + 36
Rearranging the equation:
x^2 - x^2 - 12x - 36 = 0
Simplifying the equation:
-12x - 36 = 0
Dividing both sides of the equation by -12:
x = 3
So, the length of one side of the flower bed is 3 feet.
Since the width of each side of the flower bed with the path is "x + 6" feet, the dimensions of the flower bed would be 3 feet by 3 feet.