Correct Answer: Option C Let's break down the problem step-by-step to arrive at the solution. **Step 1: Determine the properties of the cone from the sector.** When a circular sector is folded to form a cone: * The radius of the sector becomes the slant height of the cone ($l$). Given, radius of sector $R = 12\text{ cm}$. So, the slant height of the cone $l = 12\text{ cm}$. * The arc length of the sector becomes the circumference of the base of the cone. The arc length ($L$) of a sector is given by the formula $L = \frac{\theta}{360^\circ} \times 2\pi R$. Given, angle of sector $\theta = 120^\circ$. $L = \frac{120^\circ}{360^\circ} \times 2\pi (12)$ $L = \frac{1}{3} \times 24\pi$ $L = 8\pi\text{ cm}$. Let the base radius of the cone be $r$. The circumference of the cone's base is $2\pi r$. Thus, $2\pi r = 8\pi$. Dividing both sides by $2\pi$, we get $r = 4\text{ cm}$. **Step 2: Calculate the height of the cone.** For a right circular cone, the slant height ($l$), base radius ($r$), and height ($h$) are related by the Pythagorean theorem: $l^2 = r^2 + h^2$. We have $l = 12\text{ cm}$ and $r = 4\text{ cm}$. $12^2 = 4^2 + h^2$ $144 = 16 + h^2$ $h^2 = 144 - 16$ $h^2 = 128$ $h = \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}\text{ cm}$. **Step 3: Calculate the volume of the cone ($V_{\text{cone}}$).** The formula for the volume of a cone is $V_{\text{cone}} = \frac{1}{3}\pi r^2 h$. Substituting the values of $r = 4\text{ cm}$ and $h = 8\sqrt{2}\text{ cm}$: $V_{\text{cone}} = \frac{1}{3}\pi (4)^2 (8\sqrt{2})$ $V_{\text{cone}} = \frac{1}{3}\pi (16) (8\sqrt{2})$ $V_{\text{cone}} = \frac{128\sqrt{2}\pi}{3}\text{ cm}^3$. **Step 4: Calculate the volume of the cylinder ($V_{\text{cylinder}}$).** The problem states that the cylinder has the same base radius and height as the cone. So, for the cylinder: Base radius $= r_{\text{cyl}} = r_{\text{cone}} = 4\text{ cm}$. Height $= h_{\text{cyl}} = h_{\text{cone}} = 8\sqrt{2}\text{ cm}$. The formula for the volume of a cylinder is $V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 h_{\text{cyl}}$. Substituting the values: $V_{\text{cylinder}} = \pi (4)^2 (8\sqrt{2})$ $V_{\text{cylinder}} = \pi (16) (8\sqrt{2})$ $V_{\text{cylinder}} = 128\sqrt{2}\pi\text{ cm}^3$. **Step 5: Determine the ratio of the volume of the cone to the volume of the cylinder.** Ratio $= \frac{V_{\text{cone}}}{V_{\text{cylinder}}}$ Ratio $= \frac{\frac{128\sqrt{2}\pi}{3}}{128\sqrt{2}\pi}$ Ratio $= \frac{1}{3}$. Thus, the ratio of the volume of the cone to the volume of the cylinder is $1:3$. The final answer is $\boxed{C}$