Understanding Cone Volume Changes

Let's analyze how the volume of a cone changes when its dimensions are altered. The volume of a cone depends on its base radius and its height.

Cone Volume Formula

The formula for the volume of a cone is given by:

\(V = \frac{1}{3}\pi r^2 h\)

Where:

• \(V\) is the volume of the cone
• \(r\) is the base radius of the cone
• \(h\) is the height of the cone
• \(\pi\) is a mathematical constant (approximately 3.14159)

Original Cone Dimensions and Volume

Let's denote the dimensions of the original cone as \(r_1\) for the base radius and \(h_1\) for the height. The volume of the original cone, \(V_1\), is:

\(V_1 = \frac{1}{3}\pi r_1^2 h_1\)

New Cone Dimensions

According to the question, the base radius of the new cone is doubled, and its height is halved. Let the new radius be \(r_2\) and the new height be \(h_2\). So, we have:

• New radius, \(r_2 = 2 \times r_1\)
• New height, \(h_2 = \frac{1}{2} \times h_1\)

Calculating the Volume of the New Cone

Now, let's find the volume of the new cone, \(V_2\), using the new dimensions \(r_2\) and \(h_2\) in the cone volume formula:

\(V_2 = \frac{1}{3}\pi r_2^2 h_2\)

Substitute the values of \(r_2\) and \(h_2\) in terms of \(r_1\) and \(h_1\):

\(V_2 = \frac{1}{3}\pi (2r_1)^2 (\frac{1}{2}h_1)\)

Now, let's simplify the expression:

\(V_2 = \frac{1}{3}\pi (4r_1^2) (\frac{1}{2}h_1)\)

\(V_2 = \frac{1}{3}\pi \times 4 \times \frac{1}{2} \times r_1^2 h_1\)

\(V_2 = \frac{1}{3}\pi \times 2 \times r_1^2 h_1\)

\(V_2 = 2 \times (\frac{1}{3}\pi r_1^2 h_1)\)

Comparing New Volume to Original Volume

We can see that the expression in the parenthesis, \((\frac{1}{3}\pi r_1^2 h_1)\), is the volume of the original cone, \(V_1\).

So, we have:

\(V_2 = 2 \times V_1\)

This means the volume of the new cone is 2 times the volume of the original cone.

Conclusion

When the base radius of a cone is doubled and its height is halved, the volume of the new cone is 2 times the volume of the original cone.

Revision Table: Cone Volume Calculation

Additional Information: Factors Affecting Cone Volume

The volume of a cone is directly proportional to the square of its radius and directly proportional to its height. This means:

• If you double the radius (keeping height constant), the volume becomes \( (2r)^2 = 4r^2 \), so the volume becomes 4 times the original.
• If you double the height (keeping radius constant), the volume becomes \( 2h \), so the volume becomes 2 times the original.
• If you halve the radius (keeping height constant), the volume becomes \( (\frac{1}{2}r)^2 = \frac{1}{4}r^2 \), so the volume becomes 1/4 of the original.
• If you halve the height (keeping radius constant), the volume becomes \( \frac{1}{2}h \), so the volume becomes 1/2 of the original.

In this question, the radius was doubled (contributing a factor of \(2^2=4\) to the volume) and the height was halved (contributing a factor of \(\frac{1}{2}\) to the volume). The combined effect is multiplying the original volume by \(4 \times \frac{1}{2} = 2\).