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).
I can't see your diagram, but the diagonal of the square from (4,1) to (5,8) is also the diameter of the circle
length = √(1^2 + 7^2) = √50 = 5√2
so the radius of the circle must be 2.5√2
perimeter of circle = 2πr or dπ
= √50π
= appr 22.2
To calculate the amount of fencing Mabel needs to purchase, we need to find the perimeter of the circular fence surrounding the square pool.
First, let's find the length of one side of the square pool. We can use the distance formula to calculate the distance between two adjacent points.
The distance formula is:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates (1, 5) and (5, 8):
d = √((5 - 1)^2 + (8 - 5)^2)
d = √(4^2 + 3^2)
d = √(16 + 9)
d = √25
d = 5
So, the length of one side of the square pool is 5 units.
Since the square pool has four sides of equal length, the total perimeter of the pool is 4 times the length of one side:
Perimeter of square pool = 4 * 5 = 20 units.
Next, let's find the diameter of the circular fence. The diameter of the fence is equal to the diagonal of the square pool.
Again, we can use the distance formula to calculate the distance between two opposite corners of the square pool.
Using the coordinates (1, 5) and (8, 4):
d = √((8 - 1)^2 + (4 - 5)^2)
d = √(7^2 + (-1)^2)
d = √(49 + 1)
d = √50
d ≈ 7.07
So, the diameter of the circular fence is approximately 7.07 units.
The circumference of a circle is given by the formula:
Circumference = π * diameter
Using the approximate value for π as 3.14:
Circumference = 3.14 * 7.07
Circumference ≈ 22.19
Therefore, Mabel needs to purchase approximately 22.19 units of fencing to enclose the circular fence around the square pool.