https://r2.erweima.ai/mathbot/image/1724127803718xgpmq797.jpg

https://r2.erweima.ai/mathbot/image/1724127803718cxqkui5k.jpg

https://r2.erweima.ai/mathbot/image/17241278037185oladpee.jpg

Let's address each problem from the images you uploaded.

### Problem 1:
**Image:** `/mnt/data/1724127803718xgpmq797.jpg`

#### Problem Statement:
- You have a cube-shaped container with an edge of 20 cm filled with water. A metallic cube with an edge of 15 cm is placed inside, and it fully sinks to the bottom.
- The question asks for the height of the water level that was initially in the container.

#### Solution:
To solve this, we first calculate the volume of the metallic cube and the volume of water displaced by this cube.

- **Volume of the metallic cube** = \( 15^3 \) cm³ = 3375 cm³.
- The volume of the displaced water will be equal to this volume.

Next, we need to find the height of the water that corresponds to this volume in the container.

Let \( h \) be the height of the water rise in the container:
- **Volume of water rise** = \( 20 \times 20 \times h \) cm³.

We set this equal to the volume of the metallic cube:
\[ 20 \times 20 \times h = 3375 \]
\[ h = \frac{3375}{400} \]
\[ h = 8.4375 \text{ cm} \]

**Answer:** The water level rises approximately to 8.44 cm, so the correct option is **C) 8.44 cm**.

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### Problem 2:
**Image:** `/mnt/data/1724127803718cxqkui5k.jpg`

#### Problem Statement:
- You have a right triangle, with sides labeled as \( 2a \), \( x \), and \( a \).
- You are asked to determine the value of \( x \).

#### Solution:
Since this is a right triangle, we can use the Pythagorean theorem:

\[ (2a)^2 = x^2 + a^2 \]

Substituting and solving for \( x \):

\[ 4a^2 = x^2 + a^2 \]
\[ 3a^2 = x^2 \]
\[ x = \sqrt{3} \cdot a \]

Since \( x \) is given as \( x = \sqrt{3} \), the correct answer is **D) \(\sqrt{3}\)**.

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### Problem 3:
**Image:** `/mnt/data/17241278037185oladpee.jpg`

#### Problem Statement:
- A circular diagram is divided into 12 equal sectors. A small ball starts at a given point on the circle and follows a sequence of movements around the circle: 2 units in one direction, 3 units in another, and so on. The question is asking where the ball will end up.

#### Solution:
To solve this, we analyze the movement steps.

- Start at Point O.
- The movement instructions are given, with steps of "2 units clockwise," "1 unit counterclockwise," and so on.

Using the given sequence:
- From O, 2 units clockwise brings the ball to R.
- Then, 1 unit counterclockwise brings the ball to Q.
- Finally, 3 units clockwise brings the ball to S.

**Answer:** The final position of the ball is **S**.

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Would you like to go into more details on any of these problems or explore related concepts?

### Related Questions:
1. What is the principle behind the Archimedes' law of displacement?
2. How does the Pythagorean theorem apply to real-world problems?
3. Can we generalize the movement problem for any n divisions of the circle?
4. What would happen if the dimensions of the cube were different?
5. How does trigonometry apply to the second problem?

### Tip:
When solving geometric problems, always draw diagrams and label them clearly. This helps in visualizing and solving the problem more efficiently.

Explore geometry and Pythagorean theorem problems with detailed solutions. Learn how to calculate volume, apply the Pythagorean theorem, and analyze circular motion scenarios. Suitable for high school students.

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