1. **State the problem:** We have a right circular cylinder with radius $r$ and height $h$. A second cylinder has volume 392 times the volume of the first. We want to find possible radius $R$ and height $H$ of the second cylinder in terms of $r$ and $h$. 2. **Formula for volume of a cylinder:** $$V = \pi r^2 h$$ 3. **Volume of first cylinder:** $$V_1 = \pi r^2 h$$ 4. **Volume of second cylinder:** $$V_2 = 392 V_1 = 392 \pi r^2 h$$ 5. **Express $V_2$ in terms of $R$ and $H$:** $$V_2 = \pi R^2 H$$ 6. **Set volumes equal:** $$\pi R^2 H = 392 \pi r^2 h$$ 7. **Cancel $\pi$ on both sides:** $$\cancel{\pi} R^2 H = 392 \cancel{\pi} r^2 h$$ 8. **Divide both sides by $r^2 h$:** $$\frac{R^2 H}{r^2 h} = 392$$ 9. **Rewrite as:** $$\left(\frac{R}{r}\right)^2 \cdot \frac{H}{h} = 392$$ 10. **Check each option:** - (a) $R=8r$, $H=7h$: $$8^2 \times 7 = 64 \times 7 = 448 \neq 392$$ - (b) $R=8r$, $H=49h$: $$8^2 \times 49 = 64 \times 49 = 3136 \neq 392$$ - (c) $R=7r$, $H=8h$: $$7^2 \times 8 = 49 \times 8 = 392$$ (Correct) - (d) $R=49r$, $H=8h$: $$49^2 \times 8 = 2401 \times 8 = 19208 \neq 392$$ **Final answer:** Option (c) $R=7r$ and $H=8h$ satisfies the volume condition.