Cone Volume Fundamentals

A right circular cone is a three-dimensional geometric shape characterized by a circular base and a single vertex (apex). The volume of such a cone is a measure of the space it occupies and is directly related to its base radius and height. Understanding the formula for the volume of a cone is crucial for solving problems involving changes in its dimensions.

The standard formula for the volume \(V\) of a right circular cone is:

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

Where:

• \(r\) represents the radius of the circular base.
• \(h\) denotes the perpendicular height from the base to the apex.
• \(\pi\) (pi) is a mathematical constant, approximately 3.14159.

Initial Cone Volume Calculation

To determine the percentage increase in volume, we first need to establish a baseline. Let's consider the original dimensions of the right circular cone. We will denote the initial radius as \(r_1\) and the constant height as \(h\). The problem statement implies that only the radius changes, so the height remains unaffected.

Based on these initial dimensions, the initial volume of the cone, \(V_1\), can be expressed using the volume formula:

\[V_1 = \frac{1}{3}\pi r_1^2 h\]

Radius Increase and New Cone Volume

The core of the problem involves an increase in the radius. Specifically, the radius of the right circular cone is increased by 50%. This means the new radius will be the original radius plus half of the original radius.

Let the new radius be \(r_2\).

We can calculate \(r_2\) as follows:

\[r_2 = r_1 + 50\% \text{ of } r_1\]

\[r_2 = r_1 + \frac{50}{100} r_1\]

\[r_2 = r_1 + 0.5 r_1\]

\[r_2 = 1.5 r_1\]

Now, we can calculate the new volume of the cone, \(V_2\), using this updated radius \(r_2\) and keeping the height \(h\) the same:

\[V_2 = \frac{1}{3}\pi r_2^2 h\]

Substitute the expression for \(r_2\) into the formula:

\[V_2 = \frac{1}{3}\pi (1.5 r_1)^2 h\]

Square the term \((1.5 r_1)\):

\[V_2 = \frac{1}{3}\pi (2.25 r_1^2) h\]

By rearranging the terms, we can clearly see the relationship between the new volume \(V_2\) and the initial volume \(V_1\):

\[V_2 = 2.25 \times \left(\frac{1}{3}\pi r_1^2 h\right)\]

Since we know that \(V_1 = \frac{1}{3}\pi r_1^2 h\), we can substitute \(V_1\) into the equation for \(V_2\):

\[V_2 = 2.25 V_1\]

This shows that the new volume is 2.25 times the original volume.

Percentage Volume Increase Determination

To find the percentage increase in the cone's volume, we use the standard formula for percentage change:

\[\text{Percentage Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%\]

In this specific case, the original value is \(V_1\) and the new value is \(V_2\).

\[\text{Percentage Increase} = \frac{V_2 - V_1}{V_1} \times 100\%\]

Now, substitute the relationship \(V_2 = 2.25 V_1\) into the formula:

\[\text{Percentage Increase} = \frac{2.25 V_1 - V_1}{V_1} \times 100\%\]

Simplify the numerator:

\[\text{Percentage Increase} = \frac{(2.25 - 1) V_1}{V_1} \times 100\%\]

\[\text{Percentage Increase} = \frac{1.25 V_1}{V_1} \times 100\%\]

Cancel out \(V_1\) from the numerator and denominator:

\[\text{Percentage Increase} = 1.25 \times 100\%\]

\[\text{Percentage Increase} = 125\%\]

Thus, when the radius of a right circular cone is increased by 50%, its volume increases by 125%.

Summary of Volume Change

Here is a concise summary illustrating the transformation of the cone's dimensions and its impact on volume: