Final answer:
The maximum error in the calculated surface area of a sphere, when given an error in circumference measurement, can be estimated using linear approximation by first finding the error in radius and then using differentiation of the surface area formula to find the error in the area. The relative error is calculated by dividing this area error by the actual surface area.
Step-by-step explanation:
To estimate the maximum error in the calculated surface area of a sphere with a circumference measurement error, we first have to calculate the radius. The formula for the circumference C of a sphere is C = 2πr, where r is the radius. From the measured circumference of 80 cm, the radius would be r = C / (2π) = 80 cm / (2π).
Let's denote the possible error in the circumference as ∆C = 0.5 cm. The error in the radius, ∆r, is then ∆C / (2π). This error in the radius can be used to calculate the maximum error in surface area (∆A) using linear approximation, since the surface area A of a sphere is A = 4πr2.
Through differentiation, dA = 8πr dr. Therefore, the maximum error in surface area is about ∆A ≈ 8πr∆r. The relative error can be found by dividing this ∆A by the actual surface area, giving us the relative error formula ∆A / A.