Finding the Volume Ratio of Cone to Cylinder

This problem asks us to find the ratio of the volume of a cone to the volume of a cylinder, given that they have the same height and a specific relationship between their radii.

Let's break down the problem step-by-step:

1. Understand the given information about the cone and the cylinder.
2. Recall the formulas for the volume of a cone and a cylinder.
3. Substitute the given relationships into the volume formulas.
4. Calculate the ratio of the volumes.

Given Information

• The cone and the cylinder have the same height. Let's denote this height by \(h\).
• The radius of the cone is twice the radius of the cylinder. Let the radius of the cylinder be \(r_{c}\). Then the radius of the cone, \(r_{k}\), is \(2 \times r_{c}\), or \(r_{k} = 2r_{c}\).

Volume Formulas

The formula for the volume of a cone is:

\(V_{k} = \frac{1}{3} \pi r_{k}^2 h\)

The formula for the volume of a cylinder is:

\(V_{c} = \pi r_{c}^2 h\)

Calculating the Volumes with Given Relationship

Now, we substitute the relationship between the radii (\(r_{k} = 2r_{c}\)) into the volume formula for the cone:

\(V_{k} = \frac{1}{3} \pi (2r_{c})^2 h\)

Simplify the term \((2r_{c})^2\):

\((2r_{c})^2 = 2^2 \times r_{c}^2 = 4r_{c}^2\)

Substitute this back into the cone volume formula:

\(V_{k} = \frac{1}{3} \pi (4r_{c}^2) h\)

\(V_{k} = \frac{4}{3} \pi r_{c}^2 h\)

The volume of the cylinder is simply:

\(V_{c} = \pi r_{c}^2 h\)

Finding the Ratio of Volumes

The question asks for the ratio of the volume of the cone to that of the cylinder, which is \(V_{k} : V_{c}\) or \(\frac{V_{k}}{V_{c}}\).

Let's calculate the ratio:

\(\frac{V_{k}}{V_{c}} = \frac{\frac{4}{3} \pi r_{c}^2 h}{\pi r_{c}^2 h}\)

We can cancel out the common terms \(\pi r_{c}^2 h\) from the numerator and the denominator, provided that \(\pi r_{c}^2 h \neq 0\). Since radius and height are dimensions of physical objects, they are positive, so this condition is met.

\(\frac{V_{k}}{V_{c}} = \frac{\frac{4}{3}}{1}\)

\(\frac{V_{k}}{V_{c}} = \frac{4}{3}\)

So, the ratio of the volume of the cone to that of the cylinder is 4 : 3.

Comparison with Options

Let's check the given options:

Our calculated ratio is 4 : 3, which matches Option 4.

Summary of Calculation

Let \(h\) be the height and \(r_{c}\) be the radius of the cylinder. The radius of the cone is \(r_{k} = 2r_{c}\). Both have height \(h\).

Volume of cone, \(V_{k} = \frac{1}{3} \pi r_{k}^2 h = \frac{1}{3} \pi (2r_{c})^2 h = \frac{1}{3} \pi (4r_{c}^2) h = \frac{4}{3} \pi r_{c}^2 h\).

Volume of cylinder, \(V_{c} = \pi r_{c}^2 h\).

Ratio \(V_{k} : V_{c} = \frac{\frac{4}{3} \pi r_{c}^2 h}{\pi r_{c}^2 h} = \frac{4}{3}\).

The ratio is 4 : 3.

Revision Table: Cone and Cylinder Volumes

Additional Information: Solids and Volume Ratios

Understanding the formulas for the volumes of basic 3D shapes like cones, cylinders, and spheres is crucial in geometry. Problems often involve comparing the volumes of these shapes under specific conditions, such as having equal heights, equal radii, or related dimensions.

Key concepts:

• Volume: The amount of 3D space a solid occupies.
• Cone: A 3D geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex. Its volume is one-third the volume of a cylinder with the same base and height.
• Cylinder: A 3D solid with two parallel bases (usually circular) and a curved surface connecting them.
• Ratio: A comparison of two quantities. In this case, the ratio of volumes is expressed as a fraction or using a colon.

When tackling ratio problems involving geometric shapes, always:

1. Identify the shapes involved.
2. Note the given relationships between their dimensions (heights, radii, etc.).
3. Write down the volume formulas for each shape.
4. Substitute the given relationships into the formulas to express volumes in terms of common variables.
5. Form the required ratio and simplify.

This structured approach helps in solving such problems systematically and accurately.