Calculating Percentage Increase in Cone Volume

This problem asks us to find the percentage increase in the volume of a right circular cone when its radius is increased by a certain percentage, specifically p%, while its height remains unchanged. We need to use the formula for the volume of a cone and then calculate the percentage change.

Understanding the Cone Volume Formula

The volume (V) of a right circular cone is given by the formula:

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

where:

• \(r\) is the radius of the base of the cone.
• \(h\) is the height of the cone.
• \(\pi\) is a mathematical constant approximately equal to 3.14159.

Setting up the Initial and New Dimensions

Let's denote the initial dimensions and volume as follows:

• Initial radius = \(r_1\)
• Initial height = \(h_1\)
• Initial volume = \(V_1\)

So, the initial volume is \(V_1 = \frac{1}{3} \pi r_1^2 h_1\).

Now, the problem states that the radius is increased by p% without increasing the height. So, the new dimensions are:

• New radius = \(r_2\)
• New height = \(h_2\)
• New volume = \(V_2\)

According to the problem:

• \(h_2 = h_1\) (height remains the same)
• \(r_2\) is \(r_1\) increased by p%. This can be written as:

\(r_2 = r_1 + r_1 \times \frac{p}{100} = r_1 \left(1 + \frac{p}{100}\right)\)

Calculating the New Volume

Using the formula for the volume of a cone with the new radius \(r_2\) and new height \(h_2\):

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

Substitute \(r_2 = r_1 \left(1 + \frac{p}{100}\right)\) and \(h_2 = h_1\):

\(V_2 = \frac{1}{3} \pi \left(r_1 \left(1 + \frac{p}{100}\right)\right)^2 h_1\)

\(V_2 = \frac{1}{3} \pi r_1^2 \left(1 + \frac{p}{100}\right)^2 h_1\)

Determining the Increase in Volume

The increase in volume is the difference between the new volume and the initial volume:

\(\text{Increase in Volume} = V_2 - V_1\)

\(V_2 - V_1 = \frac{1}{3} \pi r_1^2 \left(1 + \frac{p}{100}\right)^2 h_1 - \frac{1}{3} \pi r_1^2 h_1\)

Factor out the common term \(\frac{1}{3} \pi r_1^2 h_1\) (which is \(V_1\)):

\(V_2 - V_1 = \frac{1}{3} \pi r_1^2 h_1 \left[ \left(1 + \frac{p}{100}\right)^2 - 1 \right]\)

\(V_2 - V_1 = V_1 \left[ \left(1 + \frac{p}{100}\right)^2 - 1 \right]\)

Calculating the Percentage Increase in Volume

The percentage increase in volume is given by the formula:

\(\text{Percentage Increase} = \left(\frac{\text{Increase in Volume}}{\text{Initial Volume}}\right) \times 100\)

\(\text{Percentage Increase} = \left(\frac{V_2 - V_1}{V_1}\right) \times 100\)

Substitute the expression for \(V_2 - V_1\):

\(\text{Percentage Increase} = \left(\frac{V_1 \left[ \left(1 + \frac{p}{100}\right)^2 - 1 \right]}{V_1}\right) \times 100\)

\(\text{Percentage Increase} = \left[ \left(1 + \frac{p}{100}\right)^2 - 1 \right] \times 100\)

Now, let's expand the term \(\left(1 + \frac{p}{100}\right)^2\) using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\):

\(\left(1 + \frac{p}{100}\right)^2 = 1^2 + 2 \times 1 \times \frac{p}{100} + \left(\frac{p}{100}\right)^2\)

\(\left(1 + \frac{p}{100}\right)^2 = 1 + \frac{2p}{100} + \frac{p^2}{10000}\)

Substitute this back into the percentage increase formula:

\(\text{Percentage Increase} = \left[ \left(1 + \frac{2p}{100} + \frac{p^2}{10000}\right) - 1 \right] \times 100\)

\(\text{Percentage Increase} = \left[ \frac{2p}{100} + \frac{p^2}{10000} \right] \times 100\)

Distribute the multiplication by 100:

\(\text{Percentage Increase} = \left(\frac{2p}{100} \times 100\right) + \left(\frac{p^2}{10000} \times 100\right)\)

\(\text{Percentage Increase} = 2p + \frac{p^2}{100}\)

This expression can be factored by taking 'p' common:

\(\text{Percentage Increase} = p \left(2 + \frac{p}{100}\right)\)

This is the percentage increase in the volume of the cone.

Comparing with Options

Let's compare our result \(p \left(2 + \frac{p}{100}\right)\) with the given options:

Our calculated percentage increase matches the expression in Option 4.

Revision Table: Cone Volume Percentage Change

Additional Information: Geometric Volume Changes

When dimensions of a geometric shape change, its volume changes based on how the dimension affects the volume formula. For a cone, volume is proportional to the square of the radius (\(r^2\)) and linearly proportional to the height (\(h\)).

• If only height changes by a percentage, the volume changes by the same percentage.
• If only radius changes by a percentage, the volume change is more complex because the radius is squared in the formula. An increase of p% in radius results in a change factor of \(\left(1 + \frac{p}{100}\right)^2\) in the \(r^2\) term.
• If both dimensions change, the calculation involves multiplying the respective change factors.

In this problem, increasing the radius by p% means the \(r^2\) term increases by a factor of \(\left(1 + \frac{p}{100}\right)^2\). Since the height is constant, the volume changes by this same factor relative to its dependence on \(r^2\), leading to the percentage increase we calculated.