To find the dimensions of the square lawn, let’s denote the length of one side of the lawn as x meters. The area of the square lawn (A_lawn) can then be calculated as:
A_lawn = x²
Since this lawn is bordered by a 4m wide path on three sides, we need to consider the overall area that includes both the lawn and the path.
The length of the side of the entire area that includes the lawn and the path can be expressed as:
Length of side of lawn + width of path on two sides = x + 4 + 4 = x + 8 meters
Hence, the area including the path (A_total) becomes:
A_total = (x + 8)²
The area of the path (A_path) can be determined by subtracting the area of the lawn from the total area:
A_path = A_total – A_lawn = (x + 8)² – x²
Next, we know that the area of the path is 78% that of the lawn area:
A_path = 0.78 * A_lawn
We can now set these equations equal to each other:
(x + 8)² – x² = 0.78 * x²
Expanding the left side:
x² + 16x + 64 – x² = 0.78x²
Now simplifying it gives:
16x + 64 = 0.78x²
Rearranging the equation leads us to:
0.78x² – 16x – 64 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a, where a = 0.78, b = -16, and c = -64.
Calculating the discriminant (b² – 4ac):
(-16)² – 4 * (0.78) * (-64) = 256 + 198.72 = 454.72
Now plugging in the values:
x = (16 ± √454.72) / (2 * 0.78)
Solving gives two results, but we will take the positive one, which can be calculated as:
x ≈ 20.13 meters
Thus, the dimensions of the lawn are:
Length of each side: approximately 20.13 meters
In conclusion, to obtain the area ratio with the path effectively considered, we find that the square lawn with a 4m wide path on three sides will have a dimension of approximately 20.13m.