A square lawn is surrounded by a path 2.5m wide, if the area of the path is 165m square, find the area of the lawn
let x be the length of each of the lawn's side, and y be the length of each of the whole land's side (including the path surrounding the lawn)
y^2 - x^2 = 165
y = x + 5
substitute y:
(x+5)^2 - x^2 = 165
hence: x^2 + 10x + 25 - x^2 = 165
10x = 140
x = 14
area of lawn: 14^2
196
To find the area of the lawn, we need to subtract the area of the path from the total area.
1. Let's assume the side length of the square lawn is "x" meters.
2. We know that the width of the path is 2.5m. Therefore, the outer side length of the square formed by the path and the lawn would be (x + 2.5 + 2.5) meters.
3. The area of the path can be calculated by subtracting the area of the lawn from the area of the outer square. So, the area of the path is (x + 2.5 + 2.5)^2 square meters.
4. Given that the area of the path is 165 square meters, we can set up the equation: (x + 2.5 + 2.5)^2 = 165.
5. Simplifying the equation: (x + 5)^2 = 165.
6. Taking the square root of both sides, we get x + 5 = √165.
7. Subtracting 5 from both sides, we have x = √165 - 5.
8. Now, to find the area of the lawn, we'll calculate the square of x: (x)^2 = (√165 - 5)^2.
9. Using a calculator, we find that x^2 ≈ 97.38 square meters.
Therefore, the area of the lawn is approximately 97.38 square meters.
To find the area of the lawn, we first need to determine the dimensions of the square. Let's assume the side length of the square lawn is "x" meters.
Since the path surrounds the lawn on all four sides, the dimensions of the larger square (including the lawn and the path) would be (x + 2.5) meters wide and (x + 2.5) meters long.
The total area of the larger square (including the path) can be calculated by multiplying the width and length:
Total Area = (x + 2.5) * (x + 2.5)
Now, let's remove the area taken up by the path from the total area to find the area of the lawn. Given that the area of the path is 165 square meters, we can subtract it from the total area:
Area of Lawn = Total Area - Area of Path
Area of Lawn = (x + 2.5) * (x + 2.5) - 165
We can simplify the equation and then solve for "x":
x^2 + 5x + 6.25 - 165 = 0
x^2 + 5x - 158.75 = 0
Now, we can solve this quadratic equation using factoring, completing the square method, or the quadratic formula. Solving for "x" will give us the side length of the lawn, which we will then use to calculate the area of the lawn.