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The text in the image reads:

"A water cylinder with the bottom end closed with a lid and with a diameter of 4 meters. The height is filled to a depth of 6.5 meters. How much will the height of the liquid change after 200 liters is withdrawn from the cylinder?"

Let’s break down the problem.

### Key information:
- The cylinder has a diameter of 4 meters, so the radius \( r \) is half of that: \( r = 2 \) meters.
- The cylinder is filled to a height of 6.5 meters.
- We need to find how much the liquid height will change when 200 liters of water is withdrawn.

### Step 1: Calculate the volume of water withdrawn
200 liters is equivalent to 0.2 cubic meters (since 1 liter = 0.001 cubic meters).

### Step 2: Volume formula for a cylinder
The volume \( V \) of a cylinder is given by:
\[
V = \pi r^2 h
\]
where:
- \( r \) is the radius,
- \( h \) is the height of the water level.

### Step 3: Find the change in height
Since the water level changes, we need to solve for the height \( \Delta h \) corresponding to a volume change of 0.2 cubic meters. Using the volume formula, the change in volume is:
\[
\Delta V = \pi r^2 \Delta h
\]
Given that \( \Delta V = 0.2 \, \text{m}^3 \) and \( r = 2 \, \text{m} \), we can solve for \( \Delta h \):
\[
0.2 = \pi (2^2) \Delta h
\]
\[
0.2 = 4\pi \Delta h
\]
\[
\Delta h = \frac{0.2}{4\pi}
\]

Now let's compute this value.The height of the liquid will decrease by approximately 0.0159 meters, or 1.59 centimeters, after 200 liters of water is withdrawn.

Would you like further details or have any other questions?

Here are five related questions to explore:
1. How would the height change if the diameter of the cylinder was larger or smaller?
2. What would the height change be if 500 liters were withdrawn instead of 200 liters?
3. How can we calculate the volume if the cylinder has a slanted side (i.e., it's not perfectly vertical)?
4. How would the problem change if the cylinder wasn't completely full to start with?
5. How does the height change relate to the cross-sectional area of the cylinder?

**Tip:** In problems involving cylinders, always start by finding the radius from the diameter, as this is crucial for calculating volumes.

A water cylinder with the bottom end closed with a lid and with a diameter of 4 meters. The height is filled to a depth of 6.5 meters. How much will the height of the liquid change after 200 liters is withdrawn from the cylinder?

Find out how the height of water changes in a cylindrical tank with a 4-meter diameter and a height of 6.5 meters when 200 liters are withdrawn. This problem involves cylindrical geometry and volume calculation, suitable for Grades 9-11.

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