Final answer:
To estimate the maximum error in the surface area of a sphere, we can use differentials. The maximum error in the surface area of the sphere is approximately 78.67 cm² when the circumference is measured to be 78 cm with a possible error of 0.5 cm.
Step-by-step explanation:
To estimate the maximum error in the surface area of a sphere, we can use differentials. The surface area of a sphere is given by the formula A = 4πr². Using differentials, we can express the change in surface area (dA) in terms of the change in radius (dr). Approximating the maximum error in the surface area (ΔA) as the differential dA, we have:
ΔA = dA = 8πrdr
Since the circumference of the sphere is given as 78 cm with a possible error of 0.5 cm, we can find the maximum error in the radius (Δr) using differentials:
Δr = dr = ΔC / (2π) = 0.5 cm / (2π) = 0.079 cm
Plugging in the values, we can calculate the maximum error in the surface area:
ΔA = 8π(39)0.079 = 78.67 cm²
Therefore, the maximum error in the surface area is approximately 78.67 cm².