circle K has an area of 64 pi units ^2. Find the exact length of the arc intercepted by a 210 degree central angle in circle K.

To find the length of the arc intercepted by a central angle in a circle, we need to know the radius of the circle and the measure of the central angle. In this case, we are given that the area of circle K is 64 pi units^2.

The formula for the area of a circle is:

Area = pi * radius^2

Since the area of circle K is given as 64 pi units^2, we can set up the equation:

64 pi = pi * radius^2

By dividing both sides of the equation by pi, we can eliminate pi:

64 = radius^2

To find the exact length of the arc, we need to determine the radius of the circle. We can do this by taking the square root of both sides of the equation:

√64 = √(radius^2)

Simplifying:

8 = radius

Now that we know the radius of circle K is 8 units, we can find the length of the arc intercepted by a 210-degree central angle using the formula:

Length of Arc = (central angle / 360 degrees) * (2 * pi * radius)

Substituting the given values:

Length of Arc = (210 degrees / 360 degrees) * (2 * pi * 8)

Simplifying:

Length of Arc = (7/12) * (16 pi)

Multiply the fractions (7/12) and (16 pi):

Length of Arc = (7/12) * (16 pi) = 112 pi / 12 = 28 pi / 3

Therefore, the exact length of the arc intercepted by the 210-degree central angle in circle K is (28 pi / 3) units.