Calculating Cone Volume Percentage Increase

We need to find the percentage increase in the volume of a circular cone when both its radius and height increase by 10%.

Cone Volume Formula

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

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

Where '$r$' is the radius and '$h$' is the height.

Applying Percentage Increase

Let the original radius be $r_0$ and the original height be $h_0$. The original volume is:

$ V_0 = \frac{1}{3} \pi r_0^2 h_0 $

The radius and height increase by 10%. So, the new radius ($r_1$) and new height ($h_1$) are:

• New radius: $r_1 = r_0 + 0.10 r_0 = 1.10 r_0$
• New height: $h_1 = h_0 + 0.10 h_0 = 1.10 h_0$

Calculating New Volume

The new volume ($V_1$) is:

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

Substitute the new radius and height:

$ V_1 = \frac{1}{3} \pi (1.10 r_0)^2 (1.10 h_0) $

$ V_1 = \frac{1}{3} \pi (1.10^2 r_0^2) (1.10 h_0) $

$ V_1 = \frac{1}{3} \pi (1.21 r_0^2) (1.10 h_0) $

$ V_1 = (1.21 \times 1.10) \times \left( \frac{1}{3} \pi r_0^2 h_0 \right) $

$ V_1 = 1.331 \times V_0 $

Determining Percentage Increase

The increase in volume is $V_1 - V_0$:

$ \text{Increase} = 1.331 V_0 - V_0 = 0.331 V_0 $

The percentage increase is calculated as:

$ \text{Percentage Increase} = \left( \frac{\text{Increase}}{V_0} \right) \times 100 $

$ \text{Percentage Increase} = \left( \frac{0.331 V_0}{V_0} \right) \times 100 $

$ \text{Percentage Increase} = 0.331 \times 100 = 33.1\% $

Conclusion

The percentage increase in the volume of the circular cone is 33.1%.