Understanding the Cylindrical Vessel Problem

The question asks us to find the capacity of a cylindrical vessel. The capacity of a vessel is essentially its volume, which tells us how much it can hold. For a cylinder, the volume depends on the radius of its base and its height. We are given the circumference of the base and the height of the cylindrical vessel.

To find the volume, we first need to determine the radius of the base using the given circumference. Then, we can use the formula for the volume of a cylinder. Finally, we need to convert the volume from cubic centimeters (cm³) to litres, as the capacity is asked in litres.

Key Formulas for Cylinders

Here are the important formulas we will use:

• Circumference of the base of a cylinder (which is a circle): \(C = 2\pi r\)
• Volume of a cylinder: \(V = \pi r^2 h\)

Where:

• \(C\) is the circumference
• \(r\) is the radius of the base
• \(h\) is the height of the cylinder
• \(\pi\) is the mathematical constant (given as \(\frac{22}{7}\))

Step 1: Calculating the Radius of the Base

We are given that the circumference of the base of the cylindrical vessel is 264 cm. We can use the circumference formula to find the radius \(r\).

\(C = 2\pi r\)

Substitute the given values:

\(264 \text{ cm} = 2 \times \frac{22}{7} \times r\)

Now, we solve for \(r\):

\(264 = \frac{44}{7} \times r\)

\(r = 264 \times \frac{7}{44}\)

We can simplify the calculation. \(264 \div 44\):

\(\frac{264}{44} = \frac{132 \times 2}{22 \times 2} = \frac{132}{22} = \frac{66 \times 2}{11 \times 2} = \frac{66}{11} = 6\)

So,

\(r = 6 \times 7\)

\(r = 42 \text{ cm}\)

The radius of the base of the cylindrical vessel is 42 cm.

Step 2: Calculating the Volume of the Cylindrical Vessel

Now that we have the radius (\(r = 42\) cm) and the height (\(h = 50\) cm), we can calculate the volume using the volume formula for a cylinder:

\(V = \pi r^2 h\)

Substitute the values:

\(V = \frac{22}{7} \times (42 \text{ cm})^2 \times 50 \text{ cm}\)

\(V = \frac{22}{7} \times (42 \text{ cm} \times 42 \text{ cm}) \times 50 \text{ cm}\)

\(V = \frac{22}{7} \times 42 \times 42 \times 50 \text{ cm}^3\)

We can simplify by canceling out the 7 with one of the 42s (\(42 \div 7 = 6\)):

\(V = 22 \times 6 \times 42 \times 50 \text{ cm}^3\)

Perform the multiplication:

\(V = 132 \times 42 \times 50 \text{ cm}^3\)

\(V = 132 \times (42 \times 50) \text{ cm}^3\)

\(V = 132 \times 2100 \text{ cm}^3\)

\(V = 277200 \text{ cm}^3\)

The volume of the cylindrical vessel is 277200 cm³.

Step 3: Converting Volume from cm³ to Litres

The capacity is usually measured in litres. We know the conversion factor:

1 litre = 1000 cm³

To convert the volume from cm³ to litres, we need to divide the volume in cm³ by 1000.

Capacity (in litres) = \(\frac{\text{Volume in } cm^3}{1000}\)

Capacity = \(\frac{277200 \text{ cm}^3}{1000 \text{ cm}^3/\text{litre}}\)

Capacity = \(277.2 \text{ litres}\)

The capacity of the cylindrical vessel is 277.2 litres.

Comparing with Options

Let's look at the given options:

1. 277.2
2. 278.4
3. 280.6
4. 267.4

Our calculated capacity is 277.2 litres, which matches Option 1.

Revision Table: Cylinder Capacity and Volume

Additional Information: Understanding Capacity

Capacity refers to the amount a container can hold. It is a measure of volume, but often expressed in units suitable for liquids or gases, such as litres or gallons. Volume is a measure of the three-dimensional space occupied by an object or enclosed by a container, typically measured in cubic units like cm³ or m³.

In the metric system, there is a direct relationship between volume in cubic units and capacity in litres:

• 1 cubic centimeter (cm³) is equal to 1 milliliter (mL).
• 1000 cubic centimeters (cm³) is equal to 1 litre (L).
• 1 cubic meter (m³) is equal to 1000 litres (L).

Understanding these conversions is crucial for solving problems involving the capacity of various shapes, including cylindrical vessels.