Volume Calculation for a Right Circular Cone

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 and $h$ is the height of the cone.

Calculating Initial Volume

Let the initial radius be $r$ and the initial height be $h$. The initial volume ($V_1$) is:

$ V_1 = \frac{1}{3} \pi r^2 h $

Calculating New Volume After Radius Increase

The radius is increased by 50%. The new radius ($r'$) is:

• Increase in radius = $50\%$ of $r = 0.50r$.
• New radius, $r' = r + 0.50r = 1.5r$.

The height ($h$) remains unchanged. The new volume ($V_2$) is:

$ V_2 = \frac{1}{3} \pi (r')^2 h $

Substitute $r' = 1.5r$:

$ V_2 = \frac{1}{3} \pi (1.5r)^2 h $

$ V_2 = \frac{1}{3} \pi (2.25 r^2) h $

$ V_2 = 2.25 \times \left( \frac{1}{3} \pi r^2 h \right) $

Since $V_1 = \frac{1}{3} \pi r^2 h$, we have:

$ V_2 = 2.25 V_1 $

Determining Percentage Volume Increase

The increase in volume is:

$ \Delta V = V_2 - V_1 = 2.25 V_1 - V_1 = 1.25 V_1 $

The percentage increase in volume is calculated as:

$ \text{Percentage Increase} = \frac{\Delta V}{V_1} \times 100\% $

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

$ \text{Percentage Increase} = 1.25 \times 100\% = 125\% $

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