The curved surface area of a right circular cone is half of another right circular cone. If the ratio of their slant heights is 2:1 and that of their volumes is 3:1, find ratio of their: (a) radii (b) heights

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.

The curved surface area of a right circular cone is half of another right circular cone. If the ratio of their slant heights is 2:1 and that of their volumes is 3:1, find ratio of their: (a) radii (b) heights.

Solve a geometry problem involving two right circular cones, where the curved surface area of one cone is half that of another. Given a 2:1 slant height ratio and 3:1 volume ratio, find the ratio of their radii and heights. Detailed explanation for students in Grades 9-11.

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