Understanding the Percentage Error in Calculating the Surface Area of a Spherical Ball

The surface area of a sphere is a crucial parameter in many scientific and engineering applications. However, the accuracy of the calculated surface area is often dependent on the accuracy of the radius measurement. This article explores the relationship between the radius error and the resultant error in calculating the surface area of a spherical ball with a radius of 7 cm.

Key Concepts and Formulas

When the radius of a sphere is given as 7 cm, the surface area (SA) can be calculated using the formula:

SA 4πr2

For a sphere with a radius of 7 cm, the surface area is:

4π(72) 4π(49) ≈ 615.75 cm2

Dimensional Analysis for Error Estimation

Dimensional analysis can help us understand how errors in the radius measurement propagate to the calculated surface area. The formula for surface area is a function of the square of the radius:

A L2

The relative error in the area (dA/A) can be expressed as:

dA/A 2 × (dL/L)

This means that a 1% error in the radius measurement results in a 2% error in the calculated surface area. Therefore, if there is an error of 4% in the radius measurement, the relative error in the surface area would be:

2 × 4% 8%

Example Calculations with Error

Lets consider the scenario where the radius measurement has an error of 4%. This error can be either positive or negative, affecting the value of the radius as follows:

For a 1.04 times radius (positive error):

SA 4π(1.04 × 7)2 ≈ 4π(72.24) ≈ 667.64 cm2

The error in the surface area calculation would be:

667.64 - 615.75 51.89 cm2

The percentage error is then calculated as:

(51.89 / 615.75) × 100% ≈ 8.42%

For a 0.96 times radius (negative error):

SA 4π(0.96 × 7)2 ≈ 4π(53.76) ≈ 554.94 cm2

The error in the surface area calculation would be:

615.75 - 554.94 60.81 cm2

The percentage error is then calculated as:

(60.81 / 615.75) × 100% ≈ 9.88%

Conclusion

In summary, the percentage error in calculating the surface area of a spherical ball is directly related to the error in the radius measurement. This relationship is particularly important in applications where precise measurements of surface area are required. Understanding and accounting for these errors can significantly improve the accuracy of your calculations.