Mabel is installing a square pool in her backyard and wants a circular fence to enclose the pool to create 4 grassy areas around the pool as show in the figure. If the pool is located at the coordinates (1, 5), (5, 8), (4, 1), and (8, 4), what is the amount of fencing Mabel needs to purchase? Round your answer to the nearest tenth. (Like 5.7 or 3.4).
Did this one yesterday.
make sure the points are the same
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To find the amount of fencing needed to enclose the pool with a circular fence, we first need to determine the diameter of the circle.
To do this, we can find the maximum distance between any two points on the pool by calculating the distance between each pair of points and selecting the largest distance.
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Now, let's calculate the distances between the four points:
1. Distance between (1, 5) and (5, 8):
d1 = sqrt((5 - 1)^2 + (8 - 5)^2) = sqrt(16 + 9) = sqrt(25) = 5
2. Distance between (5, 8) and (4, 1):
d2 = sqrt((4 - 5)^2 + (1 - 8)^2) = sqrt(1 + 49) = sqrt(50) ≈ 7.1
3. Distance between (4, 1) and (8, 4):
d3 = sqrt((8 - 4)^2 + (4 - 1)^2) = sqrt(16 + 9) = sqrt(25) = 5
4. Distance between (8, 4) and (1, 5):
d4 = sqrt((1 - 8)^2 + (5 - 4)^2) = sqrt(49 + 1) = sqrt(50) ≈ 7.1
Now, we need to find the maximum distance among these four. The maximum distance is 7.1, which will be the diameter of the circular fence.
The formula to calculate the circumference (amount of fencing needed) of a circle, given the diameter (d), is:
C = π * d
Using the value of d as 7.1, let's calculate the amount of fencing needed:
C = π * 7.1 ≈ 22.3
Therefore, Mabel needs to purchase approximately 22.3 units of fencing to enclose the pool with a circular fence.