If both the radius and height of a right circular cone are increased by 20%, its volume will be increased by

Options

A. `20%`
B. `40%`
C. `60%`
D. `72.85`

Answer

To solve the problem of how much the volume of a right circular cone increases when both the radius and height are increased by 20%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the 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 and \( h \) is the height of the cone.

2. **Define Initial Dimensions**:
Let's assume the initial radius \( r \) and height \( h \) of the cone are both 10 units (for simplicity). Thus,
\[ r = 10 \quad \text{and} \quad h = 10 \]

3. **Calculate Initial Volume**:
Using the formula, the initial volume \( V_1 \) is:
\[ V_1 = \frac{1}{3} \pi (10)^2 (10) = \frac{1}{3} \pi (1000) = \frac{1000}{3} \pi \]

4. **Increase Dimensions by 20%**:
When both the radius and height are increased by 20%, the new radius \( r' \) and new height \( h' \) are:
\[ r' = 10 + 0.2 \times 10 = 12 \quad \text{and} \quad h' = 10 + 0.2 \times 10 = 12 \]

5. **Calculate New Volume**:
The new volume \( V_2 \) is:
\[ V_2 = \frac{1}{3} \pi (12)^2 (12) = \frac{1}{3} \pi (144 \times 12) = \frac{1}{3} \pi (1728) = \frac{1728}{3} \pi \]

6. **Calculate the Increase in Volume**:
Now, we find the increase in volume:
\[ \text{Increase} = V_2 - V_1 = \left(\frac{1728}{3} \pi - \frac{1000}{3} \pi\right) = \frac{1728 - 1000}{3} \pi = \frac{728}{3} \pi \]

7. **Calculate the Percentage Increase**:
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{V_1}\right) \times 100 = \left(\frac{\frac{728}{3} \pi}{\frac{1000}{3} \pi}\right) \times 100 \]
Simplifying this gives:
\[ = \left(\frac{728}{1000}\right) \times 100 = 72.8\% \]

### Final Answer:
The volume of the cone will increase by **72.8%**.