There is a cone of height 18 cm, out of which a smaller cone (which is the top portion of the original cone) with the same vertex and vertical axis is cut out. What is the ratio of the volume of the larger (actual) cone to the smaller cone, if the height of the smaller cone is 12 cm and PS/PQ=ST/QR

Let's analyze the problem step by step.

### Step 1: Identify the given information
- The height of the original (larger) cone: \( H = 18 \) cm
- The height of the smaller cone (top portion cut out): \( h = 12 \) cm
- Both cones share the same vertex and vertical axis, and it's given that \( \frac{PS}{PQ} = \frac{ST}{QR} \). This indicates that the cones are similar.

### Step 2: Use the similarity of cones
Since the two cones are similar (they have the same vertex and vertical axis), the ratio of corresponding linear dimensions (such as height, radius) of the cones will be equal. Let:
- \( R \) be the radius of the base of the original cone.
- \( r \) be the radius of the base of the smaller cone.

From the similarity of the cones, we know:
\[
\frac{r}{R} = \frac{h}{H} = \frac{12}{18} = \frac{2}{3}
\]

### Step 3: Ratio of volumes
The volumes of the cones are related to the cube of the ratio of their corresponding linear dimensions (since volume is proportional to the cube of a linear dimension):
\[
\frac{\text{Volume of smaller cone}}{\text{Volume of larger cone}} = \left(\frac{r}{R}\right)^3 = \left(\frac{2}{3}\right)^3 = \frac{8}{27}
\]

Thus, the ratio of the volume of the larger cone to the smaller cone is:
\[
\frac{\text{Volume of larger cone}}{\text{Volume of smaller cone}} = \frac{27}{8}
\]

### Final Answer:
The ratio of the volume of the larger cone to the smaller cone is \( \boxed{\frac{27}{8}} \).

Would you like further clarification or have any other questions?

### Related Questions:
1. How do you derive the formula for the volume of a cone?
2. What is the relationship between similar figures and the ratio of their volumes?
3. Can this method be applied to other shapes, like cylinders or pyramids?
4. How does changing the height of a cone affect its volume?
5. How can the concept of similar triangles be applied in real-world problems?

### Tip:
When dealing with similar shapes, remember that the ratio of their volumes is the cube of the ratio of their corresponding linear dimensions. This concept is essential in scaling problems.

Explore the ratio of volumes between a larger cone and a smaller cone cut from it, given their heights and similarity. Understand the application of cone volumes and similarity in geometric problems. Suitable for Grades 9-12.

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