1. **State the problem:** We want to find the radius values for a cylinder with height $h=5$ cm such that its volume $V$ is greater than 100 cm³ but less than 600 cm³. 2. **Formula for volume of a cylinder:** $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Given:** Height $h = 5$ cm. 4. **Set up inequalities:** $$100 < \pi r^2 \times 5 < 600$$ 5. **Divide all parts by 5:** $$\frac{100}{5} < \pi r^2 < \frac{600}{5}$$ $$20 < \pi r^2 < 120$$ 6. **Divide all parts by $\pi$:** $$\frac{20}{\pi} < r^2 < \frac{120}{\pi}$$ 7. **Take square roots:** $$\sqrt{\frac{20}{\pi}} < r < \sqrt{\frac{120}{\pi}}$$ Calculate approximate values: $$\sqrt{\frac{20}{3.1416}} \approx \sqrt{6.366} \approx 2.52$$ $$\sqrt{\frac{120}{3.1416}} \approx \sqrt{38.197} \approx 6.18$$ 8. **Interpretation:** The radius $r$ must satisfy: $$2.52 < r < 6.18$$ 9. **Check given radius options:** - 1 cm: $1 < 2.52$ (No) - 3 cm: $2.52 < 3 < 6.18$ (Yes) - 5 cm: $2.52 < 5 < 6.18$ (Yes) - 6 cm: $2.52 < 6 < 6.18$ (Yes) - 8 cm: $8 > 6.18$ (No) **Final answer:** The radius values 3 cm, 5 cm, and 6 cm result in volumes between 100 and 600 cubic centimeters.