The radius of a spherical balloon increases from 6 cm to 10 cm when more air is pumped into it. The ratio in the surface area of the original balloon and the inflated balloon is:

Options

A. `4:5`
B. `27:125`
C. `3:5`
D. `9:25`

Answer

To find the ratio of the surface area of the original balloon to the inflated balloon, we can follow these steps:

### Step 1: Understand the formula for the surface area of a sphere
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Calculate the surface area of the original balloon
The original radius of the balloon is \( r_1 = 6 \) cm. Using the formula:
\[ A_1 = 4\pi (6)^2 = 4\pi \times 36 = 144\pi \text{ cm}^2 \]

### Step 3: Calculate the surface area of the inflated balloon
The inflated radius of the balloon is \( r_2 = 10 \) cm. Using the formula:
\[ A_2 = 4\pi (10)^2 = 4\pi \times 100 = 400\pi \text{ cm}^2 \]

### Step 4: Find the ratio of the surface areas
Now, we need to find the ratio of the surface area of the original balloon to the inflated balloon:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{144\pi}{400\pi} \]
The \( \pi \) cancels out:
\[ \text{Ratio} = \frac{144}{400} \]

### Step 5: Simplify the ratio
To simplify \( \frac{144}{400} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 16:
\[ \frac{144 \div 16}{400 \div 16} = \frac{9}{25} \]

### Final Answer
The ratio of the surface area of the original balloon to the inflated balloon is:
\[ \boxed{9:25} \]

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