1. **Problem statement:** We are given the perimeter of a quarter circle as 164.22 millimeters and need to find the radius $r$ of the quarter circle, rounded to the nearest hundredth. 2. **Formula for the perimeter of a quarter circle:** The perimeter $P$ of a quarter circle includes the curved arc plus the two radii: $$P = \text{arc length} + 2r$$ The arc length of a quarter circle is one-fourth of the circumference of a full circle: $$\text{arc length} = \frac{1}{4} \times 2\pi r = \frac{\pi r}{2}$$ So the perimeter formula becomes: $$P = \frac{\pi r}{2} + 2r$$ 3. **Substitute the given perimeter and solve for $r$:** $$164.22 = \frac{\pi r}{2} + 2r$$ 4. **Factor out $r$:** $$164.22 = r \left( \frac{\pi}{2} + 2 \right)$$ 5. **Calculate the value inside the parentheses:** $$\frac{\pi}{2} + 2 \approx \frac{3.1416}{2} + 2 = 1.5708 + 2 = 3.5708$$ 6. **Solve for $r$:** $$r = \frac{164.22}{3.5708} \approx 45.98$$ 7. **Final answer:** The radius of the quarter circle is approximately **45.98 millimeters**.