Percentage Error in Surface Area of a Sphere

To determine the percentage error in the surface area of a sphere when there is an error in measuring its radius, we need to understand the formula for the surface area of a sphere, calculate the actual surface area, calculate the surface area with the measured radius, and finally calculate the percentage error.

Formula for Surface Area of a Sphere:
The surface area (A) of a sphere can be calculated using the formula:

A = 4πr²

Where:
A is the surface area of the sphere
π is a mathematical constant approximately equal to 3.14159
r is the radius of the sphere

Calculating the Actual Surface Area:
To calculate the actual surface area, we need the actual radius of the sphere. Let's assume the actual radius is denoted by r_actual.

A_actual = 4πr_actual²

Calculating the Surface Area with Measured Radius:
Given that there is an error in measuring the radius, let's assume the measured radius is denoted by r_measured. The error in measuring the radius is given to be 2%.
Therefore, the measured radius can be calculated as:

r_measured = r_actual + (2/100) * r_actual
= r_actual + 0.02 * r_actual
= 1.02 * r_actual

Now, we can calculate the surface area with the measured radius:

A_measured = 4π(1.02 * r_actual)²
= 4π(1.0404 * r_actual²)
= 4.1616πr_actual²

Calculating the Percentage Error:
The percentage error can be calculated using the following formula:

Percentage Error = [(Measured Value - Actual Value) / Actual Value] * 100

In this case, the measured value is the surface area with the measured radius (A_measured) and the actual value is the actual surface area (A_actual).

Percentage Error = [(A_measured - A_actual) / A_actual] * 100
= [(4.1616πr_actual² - 4πr_actual²) / (4πr_actual²)] * 100
= [(0.1616πr_actual²) / (4πr_actual²)] * 100
= 0.0404 * 100
= 4.04%

Therefore, the percentage error in the surface area of the sphere, when the error in measuring its radius is 2%, is 4.04%.

4%