Ranu carries water to school in a cylindrical flask with diameter 12 cm and height 21 cm. Determine the amount of water that she can carry in the flask. (Use π = \(\frac{22}{7}\))
2376 cm3
Calculating the Volume of a Cylindrical Flask
This problem asks us to find the amount of water a cylindrical flask can hold, which is equivalent to finding the volume of the cylinder. We are given the diameter and height of the flask and a specific value for π.
Let's first list the given information:
- Diameter of the cylindrical flask = 12 cm
- Height of the cylindrical flask = 21 cm
- Value of π to use = \(\frac{22}{7}\)
The formula for the volume of a cylinder is \(V = \pi r^2 h\), where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height.
First, we need to find the radius from the given diameter. The radius is half of the diameter.
Radius \(r = \frac{\text{Diameter}}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm}\).
Now we can substitute the values of π, \(r\), and \(h\) into the volume formula:
\(V = \pi r^2 h\)
\(V = \frac{22}{7} \times (6 \text{ cm})^2 \times 21 \text{ cm}\)
\(V = \frac{22}{7} \times (36 \text{ cm}^2) \times 21 \text{ cm}\)
We can simplify the calculation by canceling out the 7 in the denominator with 21:
\(V = 22 \times 36 \text{ cm}^2 \times \frac{21}{7} \text{ cm}\)
\(V = 22 \times 36 \text{ cm}^2 \times 3 \text{ cm}\)
Now, multiply the numbers:
\(V = 22 \times (36 \times 3) \text{ cm}^3\)
\(V = 22 \times 108 \text{ cm}^3\)
To calculate \(22 \times 108\):
\(22 \times 108 = 22 \times (100 + 8)\)
\(22 \times 100 = 2200\)
\(22 \times 8 = 176\)
\(2200 + 176 = 2376\)
So, the volume of the cylindrical flask is 2376 cm3. This is the amount of water Ranu can carry in the flask.
Let's summarize the calculation steps:
| Step | Description | Calculation |
|---|---|---|
| 1 | Find the radius | \(r = \frac{12}{2} = 6\) cm |
| 2 | Use the volume formula \(V = \pi r^2 h\) | \(V = \frac{22}{7} \times 6^2 \times 21\) |
| 3 | Simplify | \(V = \frac{22}{7} \times 36 \times 21\) |
| 4 | Cancel 7 with 21 | \(V = 22 \times 36 \times 3\) |
| 5 | Calculate the product | \(V = 22 \times 108 = 2376\) |
| 6 | Final Volume | 2376 cm3 |
Therefore, the amount of water Ranu can carry is 2376 cm3.
Revision Table: Volume and Area Formulas
Understanding basic geometric formulas is crucial for solving measurement problems. Here is a quick revision of related formulas:
| Shape | Formula (Variables) | Description |
|---|---|---|
| Circle Area | \(A = \pi r^2\) (r = radius) | Area of the circular base/top of the cylinder |
| Cylinder Volume | \(V = \pi r^2 h\) (r = radius, h = height) | Space occupied by the cylinder; Base Area × Height |
| Cylinder Curved Surface Area | \(CSA = 2 \pi r h\) (r = radius, h = height) | Area of the side surface of the cylinder |
| Cylinder Total Surface Area | \(TSA = 2 \pi r (r + h)\) (r = radius, h = height) | Sum of curved surface area and areas of two circular bases |
Additional Information on Volume Calculation
Volume is a three-dimensional measurement of the space occupied by an object. It is measured in cubic units, such as cm3, m3, or in liters. When calculating the volume of a regular solid like a cylinder, prism, or cube, the general idea is often related to the area of the base multiplied by the height. For a cylinder, the base is a circle, so its area is \(\pi r^2\). Multiplying this by the height \(h\) gives the volume \(V = (\text{Area of Base}) \times \text{Height} = \pi r^2 h\). Using the correct value of π and ensuring consistent units for radius and height are important steps to get the accurate volume. In this problem, both radius and height are in centimeters, so the volume is in cubic centimeters.
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