The radius of the base of a right circular cone is increased by 15% keeping the height fixed. The volume of the cone will be increased by

Options

A. `30%`
B. `31%`
C. `32.25%`
D. `34.75%`

Answer

To solve the problem of how much the volume of a right circular cone increases when the radius is increased by 15% while keeping the height fixed, we can follow these steps:

### Step 1: Understand the formula for the volume of a cone
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 of the base and \( h \) is the height of the cone.

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

### Step 3: Calculate the new radius after the increase
The radius is increased by 15%. The new radius \( r_2 \) can be calculated as:
\[ r_2 = r + 0.15r = 1.15r \]

### Step 4: Calculate the new volume with the increased radius
Now, we can calculate the new volume \( V_2 \) using the new radius \( r_2 \):
\[ V_2 = \frac{1}{3} \pi (r_2)^2 h = \frac{1}{3} \pi (1.15r)^2 h \]
Calculating \( (1.15r)^2 \):
\[ (1.15r)^2 = 1.3225r^2 \]
Thus, the new volume becomes:
\[ V_2 = \frac{1}{3} \pi (1.3225r^2) h = \frac{1.3225}{3} \pi r^2 h \]

### Step 5: Find the increase in volume
Now we can find the increase in volume \( \Delta V \):
\[ \Delta V = V_2 - V_1 = \left(\frac{1.3225}{3} \pi r^2 h\right) - \left(\frac{1}{3} \pi r^2 h\right) \]
Factoring out \( \frac{1}{3} \pi r^2 h \):
\[ \Delta V = \frac{1}{3} \pi r^2 h (1.3225 - 1) = \frac{1}{3} \pi r^2 h (0.3225) \]

### Step 6: Calculate the percentage increase in volume
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left(\frac{\Delta V}{V_1}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left(\frac{\frac{1}{3} \pi r^2 h (0.3225)}{\frac{1}{3} \pi r^2 h}\right) \times 100 = 0.3225 \times 100 = 32.25\% \]

### Final Answer
The volume of the cone will be increased by **32.25%**.

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