Given:
Radius of the circle = 36 cm
Area of the sector = 54π cm²

To find:
Length of the corresponding arc of the sector

Solution:

Step 1: Finding the angle of the sector
The area of a sector can be calculated using the formula:
Area of sector = (θ/360) * π * r², where θ is the angle of the sector in degrees and r is the radius of the circle.

Given that the area of the sector is 54π cm² and the radius is 36 cm, we can substitute these values into the formula and solve for θ:
54π = (θ/360) * π * (36)²
Simplifying the equation:
54 = (θ/360) * 36²
54 = (θ/360) * 1296
Dividing both sides by 1296:
54/1296 = θ/360
0.0417 = θ/360
θ ≈ 0.0417 * 360
θ ≈ 15

So, the angle of the sector is approximately 15 degrees.

Step 2: Calculating the length of the arc
The length of an arc can be calculated using the formula:
Length of arc = (θ/360) * 2πr, where θ is the angle of the arc in degrees and r is the radius of the circle.

Substituting the values into the formula:
Length of arc = (15/360) * 2π * 36
Simplifying the equation:
Length of arc = (1/24) * 2π * 36
Length of arc = (1/24) * 72π
Length of arc = 3π

So, the length of the corresponding arc of the sector is 3π cm or approximately 9.42 cm.

Summary:
The length of the corresponding arc of the sector is approximately 9.42 cm. The angle of the sector was found to be 15 degrees, which was used in the formula to calculate the length of the arc.