Sierra planted a 12ft. by 10ft. flower garden in her backyard. She put a gravel walkway of uniform width around it. How wide was the walkway if she used 168ft^2 of gravel?
area including path - area inside path = area of path
area outside = (12+2w)(10+2w)
area inside = (12)(10) =120
so
path area = 120 +44w + 4 w^2 -120
= 4(w^2+11w) = 168 ft^2
w^2 + 11 w = 42
w^2 + 11 w - 42 = 0
3 and - 14
so w = 3 ft wide
To find the width of the walkway, we first need to determine the area of the flower garden and the total area including the walkway.
Step 1: Calculate the area of the flower garden.
Given that the garden is rectangular with dimensions 12ft by 10ft, you can calculate the area using the formula:
Area = Length × Width.
Area of the flower garden = 12ft × 10ft = 120ft^2.
Step 2: Calculate the total area including the walkway.
Let's assume the width of the walkway is "w" feet. In that case, the dimensions of the garden, including the walkway, would be (12ft + 2w) by (10ft + 2w):
Length of the garden including the walkway = 12ft + 2w
Width of the garden including the walkway = 10ft + 2w
Now, we can find the area of the entire garden including the walkway:
Total Area = (Length including the walkway) × (Width including the walkway)
= (12ft + 2w) × (10ft + 2w)
= 120ft^2 + 24w + 20w + 4w^2
= 4w^2 + 44w + 120ft^2
Step 3: Determine the area of the walkway.
The area of the walkway can be calculated by subtracting the area of the flower garden from the total area of the garden including the walkway:
Area of the walkway = Total Area - Area of the flower garden
= (4w^2 + 44w + 120ft^2) - 120ft^2
= 4w^2 + 44w
Step 4: Solve the equation.
Given that the area of the walkway is 168ft^2, we can set up the equation:
4w^2 + 44w = 168ft^2
Simplifying the equation, we get:
4w^2 + 44w - 168ft^2 = 0
Now, we can solve this quadratic equation to find the possible values of "w".
Using factoring or the quadratic formula, we find the solutions:
w ≈ 3.29ft or w ≈ -12.79ft
Since we are dealing with the width of the walkway, the negative solution does not make sense. Therefore, the approximate width of the walkway is w ≈ 3.29ft.