To solve this problem, we need to use the formula for the arc length of a circle. The length [tex]\( L \)[/tex] of an arc of a circle is given by:
[tex]\[ L = r \theta \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( \theta \)[/tex] is the measure of the central angle in radians.
We are given the arc length [tex]\( L = 32 \pi \)[/tex] centimeters and the central angle [tex]\( \theta = \frac{8}{9} \pi \)[/tex] radians. We need to find the radius [tex]\( r \)[/tex] of the circle.
Given the formula:
[tex]\[ 32 \pi = r \left(\frac{8}{9} \pi\right) \][/tex]
To solve for [tex]\( r \)[/tex], we can divide both sides of the equation by [tex]\( \frac{8}{9} \pi \)[/tex]:
[tex]\[ r = \frac{32 \pi}{\frac{8}{9} \pi} \][/tex]
When we simplify this,
[tex]\[ r = \frac{32 \pi}{\frac{8 \pi}{9}} \][/tex]
Recall that dividing by a fraction is equivalent to multiplying by its reciprocal, so we get:
[tex]\[ r = 32 \pi \times \frac{9}{8 \pi} \][/tex]
The [tex]\( \pi \)[/tex] terms cancel out:
[tex]\[ r = 32 \times \frac{9}{8} \][/tex]
Now, simplifying the multiplication:
[tex]\[ r = 32 \times \frac{9}{8} = 36 \][/tex]
Therefore, the radius of the circle is [tex]\( \boxed{36} \)[/tex] centimeters.
