A path of uniform width surrounds a rectangular garden that is 5m wide and 12m long. The area of the path is 168m^2. Find the width of the path.
(5+2w)(12+2w) - 5*12 = 168
Now just solve for w.
232
the width of the path is 3m
To find the width of the path, we need to subtract the area of the rectangle from the total area of the rectangle and the path together.
1. Calculate the area of the rectangle by multiplying its length by its width:
Area of the rectangle = length * width = 12 m * 5 m = 60 m^2
2. Add the area of the path to the area of the rectangle:
Total area = Area of the rectangle + Area of the path = 60 m^2 + 168 m^2 = 228 m^2
3. To find the width of the path, we need to determine the dimensions of the outer rectangle.
Let's assume the width of the path is "x" meters.
The length of the outer rectangle will be the original length plus twice the width of the path on each side:
Length of the outer rectangle = 12 m + 2x
The width of the outer rectangle will be the original width plus twice the width of the path on each side:
Width of the outer rectangle = 5 m + 2x
4. The area of the outer rectangle can be calculated by multiplying its length by its width:
Area of the outer rectangle = (Length of the outer rectangle) * (Width of the outer rectangle)
Area of the outer rectangle = (12 m + 2x) * (5 m + 2x)
5. Now, we can set up an equation to solve for "x":
(12 m + 2x) * (5 m + 2x) = Total area
(12 + 2x)(5 + 2x) = 228
6. Expand and simplify the equation:
60 + 24x + 10x + 4x^2 = 228
4x^2 + 34x + 60 = 228
7. Rearrange the equation to solve for "x" by isolating the quadratic equation:
4x^2 + 34x + 60 - 228 = 0
4x^2 + 34x - 168 = 0
8. Solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, factoring is the simplest method, so we will factor the equation:
(2x - 6)(2x + 28) = 0
9. Set each factor equal to zero and solve for "x":
2x - 6 = 0 or 2x + 28 = 0
2x = 6 or 2x = -28
x = 3 or x = -14
Since the width cannot be negative, we discard the solution x = -14.
Therefore, the width of the path is 3 meters.