Calculating the Volume of a Right Circular Cylinder

The problem asks us to find the volume of a right circular cylinder given its height and a relationship between its radius and height.

We are given the following information about the right circular cylinder:

• The height of the cylinder, \(h = 14 \, \text{cm}\).
• The radius (\(r\)) of the cylinder is four times its height (\(h\)).

First, let's determine the radius of the cylinder using the given relationship:

Radius, \(r = 4 \times h\)

Substitute the given height, \(h = 14 \, \text{cm}\):

\(r = 4 \times 14 \, \text{cm}\)

\(r = 56 \, \text{cm}\)

Now that we have both the radius (\(r\)) and the height (\(h\)), we can calculate the volume of the right circular cylinder using the formula:

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

We will use the value of \(\pi\) as \(22/7\) for this calculation, as it often yields integer results in such problems.

Substitute the values of \(r = 56 \, \text{cm}\) and \(h = 14 \, \text{cm}\) into the volume formula:

\(V = \frac{22}{7} \times (56 \, \text{cm})^2 \times 14 \, \text{cm}\)

\(V = \frac{22}{7} \times (56 \times 56) \, \text{cm}^2 \times 14 \, \text{cm}\)

\(V = \frac{22}{7} \times 3136 \, \text{cm}^2 \times 14 \, \text{cm}\)

We can simplify the calculation by dividing 14 by 7:

\(V = 22 \times 3136 \, \text{cm}^2 \times \frac{14}{7} \, \text{cm}\)

\(V = 22 \times 3136 \, \text{cm}^2 \times 2 \, \text{cm}\)

Now, multiply the numbers together:

\(V = 22 \times (3136 \times 2) \, \text{cm}^3\)

\(V = 22 \times 6272 \, \text{cm}^3\)

Let's perform the final multiplication:

\(22 \times 6272 = (20 + 2) \times 6272\)

\(= 20 \times 6272 + 2 \times 6272\)

\(= 125440 + 12544\)

\(= 137984\)

So, the volume of the cylinder is \(137984 \, \text{cm}^3\).

Let's summarise the key values:

Comparing this result with the given options, we find that the calculated volume matches one of the options.

Revision Table: Cylinder Volume Calculation Steps

Additional Information on Right Circular Cylinders

A right circular cylinder is a three-dimensional solid with two parallel circular bases of the same size, connected by a curved surface. The axis joining the centers of the bases is perpendicular to the bases. Key properties and formulas include:

• Base: A circle.
• Area of Base: \(A_{\text{base}} = \pi r^2\), where \(r\) is the radius of the base.
• Lateral Surface Area: The area of the curved surface. \(A_{\text{lateral}} = 2\pi r h\), where \(h\) is the height. This is equal to the circumference of the base multiplied by the height.
• Total Surface Area: The sum of the areas of the two bases and the lateral surface area. \(A_{\text{total}} = 2 \times A_{\text{base}} + A_{\text{lateral}} = 2\pi r^2 + 2\pi r h = 2\pi r (r + h)\).
• Volume: The space occupied by the cylinder. \(V = (\text{Area of Base}) \times \text{height} = \pi r^2 h\).

Understanding these formulas is crucial for solving problems involving cylinders.