Calculating Surface Area Increase

The surface area ($A$) of a sphere is given by the formula $A = 4\pi r^2$, where $r$ is the radius.

Step-by-Step Solution

• Initial State: Let the initial radius of the balloon be $r$. The initial surface area is $A_1 = 4\pi r^2$.
• Radius Change: The radius increases by 10%. The new radius, $r'$, is calculated as: $r' = r + 0.10r = 1.10r$
• New Surface Area: Calculate the new surface area ($A_2$) using the new radius $r'$: $A_2 = 4\pi (r')^2 = 4\pi (1.10r)^2$ $A_2 = 4\pi (1.21 r^2)$ $A_2 = 1.21 \times (4\pi r^2)$ $A_2 = 1.21 A_1$
• Percentage Increase: The increase in surface area is $A_2 - A_1 = 1.21 A_1 - A_1 = 0.21 A_1$. The percentage increase is: $ \text{Percentage Increase} = \frac{\text{Increase in Area}}{\text{Initial Area}} \times 100\% $ $ \text{Percentage Increase} = \frac{0.21 A_1}{A_1} \times 100\% $ $ \text{Percentage Increase} = 0.21 \times 100\% = 21\% $

The surface area increases by 21%.