Each of the height and base radius of a cone is increased by 100%. The percentage increase in the volume of the cone is

Options

A. `700%`
B. `400%`
C. `300%`
D. `100%`

Answer

To find the percentage increase in the volume of a cone when both the height and base radius are increased by 100%, we can follow these steps:

### Step 1: Understand the initial volume of the cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the base radius and \( h \) is the height of the cone.

### Step 2: Define the initial dimensions
Let the initial radius be \( r \) and the initial height be \( h \). Therefore, the initial volume \( V_1 \) can be expressed as:
\[ V_1 = \frac{1}{3} \pi r^2 h \]

### Step 3: Determine the new dimensions after a 100% increase
A 100% increase means that both the radius and height will double:
- New radius \( r' = 2r \)
- New height \( h' = 2h \)

### Step 4: Calculate the new volume
Using the new dimensions, the new volume \( V_2 \) is:
\[ V_2 = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi (2r)^2 (2h) \]
Calculating this gives:
\[ V_2 = \frac{1}{3} \pi (4r^2)(2h) = \frac{8}{3} \pi r^2 h \]

### Step 5: Find the increase in volume
Now, we can find the increase in volume:
\[ \text{Increase in Volume} = V_2 - V_1 = \frac{8}{3} \pi r^2 h - \frac{1}{3} \pi r^2 h \]
This simplifies to:
\[ \text{Increase in Volume} = \frac{7}{3} \pi r^2 h \]

### Step 6: Calculate the percentage increase
The percentage increase in volume is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase in Volume}}{\text{Original Volume}}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{\frac{7}{3} \pi r^2 h}{\frac{1}{3} \pi r^2 h}\right) \times 100 \]
This simplifies to:
\[ \text{Percentage Increase} = \left(7\right) \times 100 = 700\% \]

### Final Answer
The percentage increase in the volume of the cone is **700%**.

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