Question

The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.

Answer 1

We are given the following information:

• Initial radius of the balloon, \( r_1 = 7 \, \text{cm} \)
• Final radius of the balloon, \( r_2 = 14 \, \text{cm} \)

We are required to find the ratio of the surface areas of the balloon in the two cases.

Step-by-Step Solution:

1. 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.

2. Surface Area in the First Case:

The surface area of the balloon when the radius is \( r_1 = 7 \, \text{cm} \) is: \[ A_1 = 4 \pi (7)^2 = 4 \pi \times 49 = 196 \pi \, \text{cm}^2 \]

3. Surface Area in the Second Case:

The surface area of the balloon when the radius is \( r_2 = 14 \, \text{cm} \) is: \[ A_2 = 4 \pi (14)^2 = 4 \pi \times 196 = 784 \pi \, \text{cm}^2 \]

4. Finding the Ratio of the Surface Areas:

The ratio of the surface areas \( \frac{A_2}{A_1} \) is: \[ \frac{A_2}{A_1} = \frac{784 \pi}{196 \pi} = \frac{784}{196} = 4 \]

Final Answer:

Therefore, the ratio of the surface areas of the balloon in the two cases is: \[ \boxed{4} \]