The image you uploaded contains the following mathematical problem:

Problem Statement:

A vessel in the form of an inverted cone is filled with water to the brim. The height of the cone is 24 cm, and the diameter is 18 cm. Some water is removed such that the height of the remaining water is half the original height. What is the volume of water that is removed?

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Let's break down the solution step by step.

Step 1: Determine the volume of the cone when full.

The formula for the volume of a cone is given by:

$V = \frac{1}{3} \pi r^2 h$

Where:

• $r$ is the radius of the base of the cone.
• $h$ is the height of the cone.

Given:

• Height $h = 24$ cm
• Diameter $d = 18$ cm, so the radius $r = \frac{d}{2} = 9$ cm

Substituting these values into the formula:

$V = \frac{1}{3} \pi (9)^2 (24)$

$V = \frac{1}{3} \pi (81) (24)$

$V = \frac{1}{3} \pi (1944)$

$V = 648 \pi \text{ cubic cm}$

Step 2: Determine the volume of water when the height is halved.

When the height is halved, the new height $h'$ is:

$h' = \frac{24}{2} = 12 \text{ cm}$

Since the shape is still a cone and the dimensions scale proportionally, the radius of the water's surface will also be halved:

$r' = \frac{9}{2} = 4.5 \text{ cm}$

Now, calculate the volume of the water remaining:

$V' = \frac{1}{3} \pi (4.5)^2 (12)$

$V' = \frac{1}{3} \pi (20.25) (12)$

$V' = \frac{1}{3} \pi (243)$

$V' = 81 \pi \text{ cubic cm}$

Step 3: Calculate the volume of water removed.

The volume of water removed is the difference between the original volume and the volume when the height is halved:

$V_{\text{removed}} = V - V' = 648 \pi - 81 \pi = 567 \pi \text{ cubic cm}$

Final Answer:

The volume of water that is removed is $567 \pi$ cubic cm. In numerical form, this is approximately $1780.972$ cubic cm.

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Would you like more details or have any questions? Here are five related questions to further your understanding:

1. How does the volume change if the cone's diameter is doubled but the height remains the same?
2. What happens to the volume if instead of halving the height, it is reduced to one-third?
3. Can you derive the relationship between the height and the volume for a cone?
4. What is the formula for the surface area of a cone and how does it relate to the volume?
5. If the cone was instead filled to a quarter of its height, what would be the volume of water?

Tip: Always remember that when scaling a 3D shape proportionally, the volume scales by the cube of the scaling factor.