The radius as well as the height of a circular cone increases by 10%. The percentage increase in its volume is ______.

Understanding the Cone Volume Problem

The question asks us to find the percentage increase in the volume of a circular cone when both its radius and height are increased by 10%. We need to calculate how this change affects the total volume.

Formula for Cone Volume

First, let's recall the formula for the volume ($V$) of a circular cone:

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

where:

- $r$ is the radius of the base
- $h$ is the height of the cone
- $\pi$ is a mathematical constant (approximately 3.14159)

Calculating the Percentage Increase

Let's denote the original radius as $r_1$ and the original height as $h_1$. The original volume $V_1$ is:

$$V_1=\frac{1}{3}\pi r_1^2{h}_1$$

The problem states that both the radius and the height increase by 10%. This means the new radius ($r_2$) and the new height ($h_2$) can be calculated as follows:

- **New Radius ($r_2$):** An increase of 10% means the new radius is the original radius plus 10% of the original radius. $$r_2=r_1+10\%\text{ of }{r}_1=r_1+0.10r_1=1.10r_1$$
- **New Height ($h_2$):** Similarly, the new height is the original height plus 10% of the original height. $$h_2=h_1+10\%\text{ of }{h}_1=h_1+0.10h_1=1.10h_1$$

Now, let's calculate the new volume ($V_2$) using the new radius ($r_2$) and new height ($h_2$):

$$V_2=\frac{1}{3}\pi r_2^2{h}_2$$

Substitute the expressions for $r_2$ and $h_2$:

$$V_2=\frac{1}{3}\pi (1.10r_1{)}^2(1.10h_1)$$

Simplify the expression:

$$V_2=\frac{1}{3}\pi (1.21r_1^2)(1.10h_1)$$

$$V_2=\frac{1}{3}\pi r_1^2{h}_1\times (1.21\times 1.10)$$

Recognize that $\frac{1}{3}\pi r_1^2{h}_1$ is the original volume $V_1$. So:

$$V_2=V_1\times (1.21\times 1.10)$$

Calculate the product $1.21\times 1.10$:

$$1.21\times 1.10=1.331$$

Therefore, the new volume is:

$$V_2=V_1\times 1.331$$

Calculating the Percentage Change in Volume

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

$$\text{Increase in Volume}=V_2-V_1$$

$$\text{Increase in Volume}=1.331V_1-V_1=(1.331-1)V_1=0.331V_1$$

To find the percentage increase, we divide the increase in volume by the original volume and multiply by 100%:

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

$$\text{Percentage Increase}=\frac{0.331V_1}{V_1}\times 100\%$$

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

Conclusion

The percentage increase in the volume of the circular cone when both the radius and height increase by 10% is 33.1%. This corresponds to option 3.