We are given two right circular cones with specific relationships:

1. The curved surface area (CSA) of the first cone is half of that of the second cone.
2. The ratio of their slant heights is 2:1.
3. The ratio of their volumes is 3:1.

We are asked to find the ratio of their radii and heights.

Key formulas:

• Curved Surface Area (CSA) of a cone:
  $\text{CSA} = \pi r l$
  where $r$ is the radius and $l$ is the slant height.

• Volume of a cone:
  $V = \frac{1}{3} \pi r^2 h$
  where $r$ is the radius and $h$ is the height.

Let’s denote:

• $r_1$ and $r_2$ as the radii of the two cones,
• $l_1$ and $l_2$ as the slant heights of the two cones,
• $h_1$ and $h_2$ as the heights of the two cones.

Step 1: Using the ratio of the curved surface areas

We know: $\text{CSA of cone 1} = \frac{1}{2} \times \text{CSA of cone 2}$ Thus: $\pi r_1 l_1 = \frac{1}{2} \times \pi r_2 l_2$ Simplifying: $r_1 l_1 = \frac{1}{2} r_2 l_2$ We are also given that the ratio of slant heights is $l_1 : l_2 = 2:1$, i.e., $l_1 = 2l_2$.

Substituting $l_1 = 2l_2$ into the equation: $r_1 \times 2l_2 = \frac{1}{2} r_2 \times l_2$ Canceling $l_2$ from both sides: $r_1 \times 2 = \frac{1}{2} r_2$ Thus: $r_1 = \frac{r_2}{4}$ So, the ratio of the radii is: $r_1 : r_2 = 1:4$

Step 2: Using the ratio of the volumes

We know: $\frac{V_1}{V_2} = \frac{3}{1}$ That is: $\frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{3}{1}$ Simplifying: $\frac{r_1^2 h_1}{r_2^2 h_2} = 3$ Substituting $r_1 = \frac{r_2}{4}$: $\frac{\left(\frac{r_2}{4}\right)^2 h_1}{r_2^2 h_2} = 3$ Simplifying: $\frac{\frac{r_2^2}{16} h_1}{r_2^2 h_2} = 3$ Canceling $r_2^2$ from both sides: $\frac{h_1}{16 h_2} = 3$ Thus: $h_1 = 48 h_2$ So, the ratio of the heights is: $h_1 : h_2 = 48:1$

Final Answer:

(a) The ratio of their radii is $1:4$.
(b) The ratio of their heights is $48:1$.

Would you like further details on these calculations or additional questions?

Here are 5 related questions to expand your understanding:

1. How does changing the radius affect the curved surface area of a cone?
2. If the height of a cone doubles, how is its volume affected?
3. How would the ratio of slant heights affect the volume ratio of two cones?
4. Can you derive a relationship between the slant height and the height of a cone?
5. How is the lateral area of a cone related to its radius and height?

Tip: When comparing geometric shapes, understanding the relation between different dimensions like height, radius, and slant height can simplify your approach to problem-solving.