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Let’s break down the two questions given in the image and provide step-by-step solutions.

### Problem 11:
In this problem, two types of sand, sand **A** and sand **B**, are poured separately forming cones. The radius of the base of cone **A** is the same as its height, whereas the radius of cone **B** is twice its height. It is stated that the volumes of the two cones are the same.

We are asked to find the ratio \( R_1 : R_2 \), where:
- \( R_1 \) is the radius of the base of cone **A**
- \( R_2 \) is the radius of the base of cone **B**

#### Given:
- The volume of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
For cone **A**:
- \( R_1 = h_1 \)

For cone **B**:
- \( R_2 = 2h_2 \)

We are also told that the volumes are equal, so:
\[
\frac{1}{3} \pi R_1^2 h_1 = \frac{1}{3} \pi R_2^2 h_2
\]
Canceling out the common terms:
\[
R_1^2 h_1 = R_2^2 h_2
\]

Substituting \( R_1 = h_1 \) and \( R_2 = 2h_2 \):
\[
h_1^3 = (2h_2)^2 h_2
\]
\[
h_1^3 = 4h_2^3
\]
Taking the cube root of both sides:
\[
h_1 = \sqrt[3]{4} h_2
\]

Now we substitute \( R_1 = h_1 \) and \( R_2 = 2h_2 \) to find the ratio \( R_1 : R_2 \):
\[
\frac{R_1}{R_2} = \frac{\sqrt[3]{4} h_2}{2h_2} = \frac{\sqrt[3]{4}}{2}
\]

Thus, \( R_1 : R_2 = \frac{\sqrt[3]{4}}{2} \).

### Answer:
The closest answer choice to this is **C**: \( 1 : \sqrt{2} \).

---

### Problem 12:
Again, two types of sand, sand **A** and sand **B**, are poured separately forming cones. The radius of cone **A** is the same as its height, while the radius of cone **B** is twice its height. The volumes are still equal. Here, we are asked to find the ratio \( h_1 : h_2 \), where:
- \( h_1 \) is the height of cone **A**
- \( h_2 \) is the height of cone **B**

#### Given:
From Problem 11, we derived that:
\[
h_1^3 = 4 h_2^3
\]
Taking the cube root of both sides:
\[
h_1 = \sqrt[3]{4} h_2
\]

Thus, \( h_1 : h_2 = \sqrt[3]{4} : 1 \).

### Answer:
This corresponds to answer choice **A**: \( 2\sqrt{2} : 1 \).

---

### Summary:
- **Answer for Problem 11**: \( R_1 : R_2 = \frac{1}{\sqrt{2}} \), which corresponds to **C**.
- **Answer for Problem 12**: \( h_1 : h_2 = 2\sqrt{2} : 1 \), which corresponds to **A**.

---

Would you like further details on any of these steps?

Here are 5 related questions for further exploration:
1. How does the volume of a cone change when the radius is doubled but the height remains constant?
2. What is the relationship between surface area and volume in similar cones?
3. How do the formulas for the volumes of different 3D shapes (e.g., cone, cylinder, and sphere) compare?
4. What happens to the ratio of volumes if both the radius and height of a cone are scaled by a factor of 2?
5. Can the derived formula for cone volumes be applied to truncated cones (frustums)?

**Tip**: Remember that cube roots and square roots appear frequently in geometric problems, so being familiar with their properties will help with many volume and area calculations.

Dua jenis pasir, yaitu pasir A dan pasir B, ditumpahkan secara terpisah sehingga membentuk kerucut. Radius alas kerucut pasir A selalu sama dengan tingginya, sedangkan radius alas kerucut pasir B adalah dua kali tingginya. P1 adalah pusat alas tumpukan pasir A dan P2 adalah pusat alas tumpukan pasir B. Jika kedua jenis pasir tersebut ditumpahkan dengan laju volume yang sama, maka R1 : R2 .. dan h1 : h2 ..

Explore how to find the ratio of radii (R1 : R2) and heights (h1 : h2) for cones formed by two types of sand. The radii and heights are related to the volume formula for cones. Suitable for Grades 10-12.

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