the rectangular garden of 6 metres by 4 metres is surrounded by a path of uniform width. if the area path is 56m squad, what is its width
L = 6 + 2 p
W = 4 + 2 p
56 = (6+2p)(4+2p)-6*4
56 = 24 + 20p + 4p^2 -24
4 p^2 + 20 p - 56 = 0
p^2 + 5 p - 14 = 0
(p+7)(p-2) = 0
p = 2
the path is 2m
To find the width of the path surrounding the rectangular garden, we can follow these steps:
1. Determine the total area of the entire plot, including the garden and the path:
Total Area = Area of Garden + Area of Path
2. Calculate the area of the rectangle garden:
Area of Garden = Length of Garden × Width of Garden
3. Calculate the area of the entire plot:
Total Area = (Length of Garden + 2 × Width of Path) × (Width of Garden + 2 × Width of Path)
4. Set up an equation using the given information:
Total Area = Area of Garden + Area of Path
(Length of Garden + 2 × Width of Path) × (Width of Garden + 2 × Width of Path) = Area of Garden + 56
5. Substitute the values into the equation:
(6 + 2w) × (4 + 2w) = 24 + 56
(6w + 12) × (4w + 8) = 80
(6w + 12) × (4w + 8) - 80 = 0
6. Solve the quadratic equation:
Simplify and rearrange the equation to form a quadratic equation:
24w^2 + 72w + 96 - 80 = 0
24w^2 + 72w + 16 = 0
Factor the equation:
(4w + 2)(6w + 8) = 0
Apply the zero-product property:
4w + 2 = 0 or 6w + 8 = 0
Solve for w:
For 4w + 2 = 0:
4w = -2
w = -2/4
w = -1/2
For 6w + 8 = 0:
6w = -8
w = -8/6
w = -4/3
Since it doesn't make sense to have a negative width, we discard the negative values.
Therefore, the width of the path surrounding the rectangular garden is 2/3 meters or approximately 0.67 meters.