To find the diameter of the exercise ball based on its surface area, we can use the formula for the surface area of a sphere:
\[ A = 4 \pi r^2 \]
Where:
- \( A \) is the surface area,
- \( r \) is the radius,
- \( \pi \) is approximately 3.14.
We're given that the surface area \( A = 13,266.5 , \text{cm}^2 \). We need to solve for the radius \( r \) first and then find the diameter \( d \) using the relationship \( d = 2r \).
- Substitute the known values into the surface area formula:
\[ 13,266.5 = 4 \times 3.14 \times r^2 \]
- Calculate \( 4 \times 3.14 \):
\[ 4 \times 3.14 = 12.56 \]
Thus, the equation simplifies to:
\[ 13,266.5 = 12.56 \times r^2 \]
- Now, solve for \( r^2 \):
\[ r^2 = \frac{13,266.5}{12.56} \]
Calculating the right-hand side:
\[ r^2 \approx \frac{13,266.5}{12.56} \approx 1,056.0 \]
- Take the square root to find \( r \):
\[ r \approx \sqrt{1,056.0} \approx 32.5 , \text{cm} \]
- Finally, calculate the diameter \( d \):
\[ d = 2r = 2 \times 32.5 = 65.0 , \text{cm} \]
Hence, the diameter of the exercise ball is approximately 65.0 cm.