The correct answer is C. Let $r_c$ and $h_c$ be the radius and height of the cone, respectively. Let $r_{cy}$ and $h_{cy}$ be the radius and height of the cylinder, respectively. From the problem statement, we are given the following conditions: 1. The height of the cone is equal to the diameter of its base. This implies $h_c = 2r_c$. 2. The cylinder has the same base radius as the cone. This implies $r_{cy} = r_c$. Let us denote this common radius as $r$. Therefore, $r_c = r$ and $r_{cy} = r$. From condition (1), substituting $r_c = r$, we get $h_c = 2r$. 3. The volume of the cylinder is four times the volume of the cone. This can be written as $V_{cy} = 4 \times V_c$. Now, let's write down the standard formulae for the volumes of a right circular cone and a right circular cylinder: Volume of a cone, $V_c = \frac{1}{3} \pi r_c^2 h_c$. Volume of a cylinder, $V_{cy} = \pi r_{cy}^2 h_{cy}$. Substitute the given conditions into these volume formulae: For the cone: Using $r_c = r$ and $h_c = 2r$, we get: $V_c = \frac{1}{3} \pi (r)^2 (2r)$ $V_c = \frac{2}{3} \pi r^3$ For the cylinder: Using $r_{cy} = r$, we get: $V_{cy} = \pi (r)^2 h_{cy}$ $V_{cy} = \pi r^2 h_{cy}$ Now, we use the relationship between their volumes as stated in condition (3): $V_{cy} = 4 \times V_c$ Substitute the expressions for $V_{cy}$ and $V_c$: $\pi r^2 h_{cy} = 4 \times \left( \frac{2}{3} \pi r^3 \right)$ $\pi r^2 h_{cy} = \frac{8}{3} \pi r^3$ To find $h_{cy}$, we can divide both sides of the equation by $\pi r^2$ (since $r$ is a radius, $r \neq 0$): $h_{cy} = \frac{8}{3} r$ The problem asks for the ratio of the height of the cylinder to the height of the cone, which is $\frac{h_{cy}}{h_c}$. We have $h_{cy} = \frac{8}{3} r$ and we found earlier that $h_c = 2r$. Now, compute the ratio: $\frac{h_{cy}}{h_c} = \frac{\frac{8}{3} r}{2r}$ Cancel out $r$ from the numerator and denominator: $\frac{h_{cy}}{h_c} = \frac{8}{3 \times 2}$ $\frac{h_{cy}}{h_c} = \frac{8}{6}$ $\frac{h_{cy}}{h_c} = \frac{4}{3}$ Thus, the ratio of the height of the cylinder to the height of the cone is $4:3$.