If the base radius and the height of a right circular cone are increased by 40% then the percentage increase in volume ( approx ) is :

Options

A. `175%`
B. `120%`
C. `64%`
D. `540%`

Answer

To find the percentage increase in the volume of a right circular cone when both the base radius and height are increased by 40%, we can follow these steps:

### Step-by-Step Solution:

1. **Define Initial Parameters:**
- Let the initial radius of the cone be \( r \).
- Let the initial height of the cone be \( h \).

2. **Calculate Initial Volume:**
- The formula for the volume \( V \) of a right circular cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]

3. **Calculate New Radius and Height:**
- Since the radius is increased by 40%, the new radius \( r' \) will be:
\[ r' = r + 0.4r = 1.4r \]
- Similarly, the height is also increased by 40%, so the new height \( h' \) will be:
\[ h' = h + 0.4h = 1.4h \]

4. **Calculate New Volume:**
- The new volume \( V' \) with the increased radius and height will be:
\[ V' = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi (1.4r)^2 (1.4h) \]
- Simplifying this gives:
\[ V' = \frac{1}{3} \pi (1.96r^2)(1.4h) = \frac{1}{3} \pi (2.744r^2h) \]

5. **Calculate the Increase in Volume:**
- The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V' - V = \left(\frac{1}{3} \pi (2.744r^2h)\right) - \left(\frac{1}{3} \pi r^2h\right) \]
- Factoring out the common term:
\[ \Delta V = \frac{1}{3} \pi r^2h (2.744 - 1) = \frac{1}{3} \pi r^2h (1.744) \]

6. **Calculate Percentage Increase in Volume:**
- The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left(\frac{\Delta V}{V}\right) \times 100 = \left(\frac{\frac{1}{3} \pi r^2h (1.744)}{\frac{1}{3} \pi r^2h}\right) \times 100 \]
- Simplifying this gives:
\[ \text{Percentage Increase} = 1.744 \times 100 \approx 174.4\% \]
- Rounding this gives approximately:
\[ \text{Percentage Increase} \approx 175\% \]

### Final Answer:
The approximate percentage increase in volume is **175%**.