Final answer:
To estimate the maximum error in the surface area and volume of a sphere using differentials, we need to find the error in the radius from the circumference error and apply it to the formulas for surface area and volume. The relative error for surface area is obtained by dividing the absolute error by the actual surface area.
Step-by-step explanation:
The question asks us to use differentials to estimate the maximum error in the surface area and volume of a sphere, given the measured circumference with a possible error.
Part A: Surface Area Error Estimation
Given:
Circumference of the sphere: C = 80 cm
Possible error in circumference: dC = ±0.5 cm
To find the radius, we use the formula: C=2πr, where r is the radius and π (pi) is a constant (~3.14159). Solving for r gives us r = C / (2π). The surface area of a sphere formula is A = 4πr². Applying differentials, we get dA = 8πr dr. Since dr is the error in the radius, and dC is 0.5 cm, we can use dC = 2πdr to find dr, and consequently dA. Plugging in the values, we can calculate dA and round to the nearest integer to find the maximum error in the surface area.
Part B: Volume Error Estimation
The volume of a sphere is given by the formula V = 4/3 πr³. Using differentials, dV = 4πr² dr. From the error in the radius (dr), we can estimate the maximum error in the volume (dV).
Relative Error
The relative error in the surface area is computed by dividing the absolute error by the actual surface area (dA/A).