If the area of the base of a cone is increased then it becomes 1.96 times of original area. The volume is increased by:

Options

A. `141%`
B. `40%`
C. `96%`
D. `100%`

Answer

To solve the problem step by step, we will follow the given information about the cone's base area and volume.

### Step 1: Understand the Area of the Base of the Cone
The area \( A \) of the base of a cone is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the base.

### Step 2: Calculate the Original Area
Let's assume the original radius \( r \) is 10 units. Then the original area \( A_1 \) will be:
\[ A_1 = \pi (10)^2 = 100\pi \]

### Step 3: Calculate the New Area
According to the problem, the area of the base is increased to 1.96 times the original area. Therefore, the new area \( A_2 \) will be:
\[ A_2 = 1.96 \times A_1 = 1.96 \times 100\pi = 196\pi \]

### Step 4: Find the New Radius
Since the area of the base is also given by \( A = \pi r^2 \), we can set the new area equal to the formula:
\[ 196\pi = \pi r_1^2 \]
Dividing both sides by \( \pi \):
\[ 196 = r_1^2 \]
Taking the square root of both sides:
\[ r_1 = \sqrt{196} = 14 \text{ units} \]

### Step 5: Calculate the Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( h \) is the height of the cone.

#### Original Volume
Using the original radius \( r = 10 \):
\[ V_1 = \frac{1}{3} \pi (10)^2 h = \frac{1}{3} \pi (100) h = \frac{100\pi h}{3} \]

#### New Volume
Using the new radius \( r_1 = 14 \):
\[ V_2 = \frac{1}{3} \pi (14)^2 h = \frac{1}{3} \pi (196) h = \frac{196\pi h}{3} \]

### Step 6: Calculate the Increase in Volume
The increase in volume \( \Delta V \) is given by:
\[ \Delta V = V_2 - V_1 = \frac{196\pi h}{3} - \frac{100\pi h}{3} = \frac{(196 - 100)\pi h}{3} = \frac{96\pi h}{3} \]

### Step 7: 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:
\[ \text{Percentage Increase} = \left(\frac{\frac{96\pi h}{3}}{\frac{100\pi h}{3}}\right) \times 100 = \left(\frac{96}{100}\right) \times 100 = 96\% \]

### Final Answer
The volume of the cone is increased by **96%**.

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