The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratios of the surfcae area of the balloom in the two cases is

Options

A. `1:4`
B. `1:3`
C. `2:3`
D. `2:1`

Explanation

To find the ratio of the surface areas of a hemispherical balloon when the radius increases from 6 cm to 12 cm, we will follow these steps:

### Step 1: Understand the formula for the surface area of a hemisphere
The surface area \( A \) of a hemisphere is given by the formula:
\[ A = 3\pi r^2 \]
where \( r \) is the radius of the hemisphere.

### Step 2: Calculate the surface area for the first radius
Let the first radius \( r_1 = 6 \) cm. Using the formula:
\[ A_1 = 3\pi (r_1)^2 = 3\pi (6)^2 = 3\pi \times 36 = 108\pi \, \text{cm}^2 \]

### Step 3: Calculate the surface area for the second radius
Now, let the second radius \( r_2 = 12 \) cm. Using the formula:
\[ A_2 = 3\pi (r_2)^2 = 3\pi (12)^2 = 3\pi \times 144 = 432\pi \, \text{cm}^2 \]

### Step 4: Find the ratio of the two surface areas
Now, we need to find the ratio of the surface areas \( A_1 \) and \( A_2 \):
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{108\pi}{432\pi} \]
The \( \pi \) cancels out:
\[ \text{Ratio} = \frac{108}{432} \]

### Step 5: Simplify the ratio
To simplify \( \frac{108}{432} \):
\[ \frac{108}{432} = \frac{1}{4} \]
Thus, the ratio of the surface areas is:
\[ 1 : 4 \]

### Final Answer
The ratio of the surface areas of the balloon in the two cases is \( 1 : 4 \).

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