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so what's the correct answer and give the way to solve it

To solve this, we need to calculate the change in the water height after adding 6 iron balls into a tube of water. The given conditions are as follows:

1. The tube initially contains 30 cm of water.
2. The diameter of the tube is 28 cm.
3. We have 6 iron balls, and the problem asks for the change in the water height for different radii of the iron balls (given in the options).

### Key formula:
The volume displaced by the iron balls will increase the water level in the tube. 

**Volume of each iron ball:**  
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the iron ball.

**Volume displaced by 6 iron balls:**
If the radius of each ball is \( r \), then the total volume displaced by 6 balls is:
\[
V_{\text{total}} = 6 \times \frac{4}{3} \pi r^3 = 8 \pi r^3
\]

**Height increase due to displacement:**
The volume displaced will increase the height of the water. The height increase \( h \) is related to the volume displaced and the cross-sectional area of the tube.

The cross-sectional area \( A \) of the tube is given by:
\[
A = \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{28}{2} \right)^2 = \pi \times 14^2 = 196\pi \, \text{cm}^2
\]

So, the height increase is:
\[
h = \frac{V_{\text{total}}}{A} = \frac{8 \pi r^3}{196 \pi} = \frac{4r^3}{98} = \frac{2r^3}{49}
\]

### Let's compute the height increase for different radii of the iron balls:

#### Option B: Radius = 7 cm
\[
h = \frac{2(7^3)}{49} = \frac{2(343)}{49} = \frac{686}{49} = 14 \, \text{cm}
\]

Total water height after the iron balls are placed:
\[
30 \, \text{cm} + 14 \, \text{cm} = 44 \, \text{cm}
\]

#### Option C: Radius = 7 cm
\[
h = 27.104 \, \text{cm} \quad (\text{incorrect as it doesn't match our computation})
\]

#### Option D: Radius = 14 cm
For radius 14 cm, we get:
\[
h = \frac{2(14^3)}{49} = \frac{2(2744)}{49} = \frac{5488}{49} \approx 112 \, \text{cm}
\]

So, total water height:
\[
30 \, \text{cm} + 112 \, \text{cm} = 142 \, \text{cm}
\]

#### Conclusion:
The correct answer is **Option D**: If the radius of each iron ball is 14 cm, the total height of water in the tube after placing 6 balls is **142 cm**. 

Would you like more details or have further questions?

### Follow-up Questions:
1. What is the relation between the volume of a sphere and its radius?
2. How would the result change if the diameter of the tube was different?
3. What happens to the height of the water if fewer iron balls are inserted?
4. How would you calculate the total volume of water displaced with 10 iron balls?
5. Why is the cross-sectional area important when calculating the height increase?

### Tip:
When solving volume displacement problems, always ensure you calculate the cross-sectional area correctly, as it directly affects the height increase!

Learn how to calculate the increase in water height after placing 6 iron balls into a tube of water. Using geometry, the volume of a sphere, and cross-sectional area concepts, we determine the correct water level for different ball radii.

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