The radius of a sphere was measured to be 10 cm with a possible error of 0.5 cm.

Question

a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer. The surface area of a sphere is S=4𝜋r².)

What is the relative error? (Round your answer to three decimal places.)

(b) Use differentials to estimate the maximum error in the calculated volume. (Round your answer to the nearest integer. The volume of a sphere is V=4/3𝜋r³.)

What is the relative error? (Round your answer to three decimal places.)

Answer 1

We are given that the radius r of a sphere was measured to be

r = (10.0 ± 0.5) cm = (r₀ + δr).

We wish to propagate the measured uncertainty in r to find the uncertainty in calculated quantity A = Cr² where C is a constant. The basic concept here is to expand the function A as a power (Taylor) series about the mean value r₀. When the mean value is much larger in magnitude than the uncertainty, |r₀| >> δr, the dominant contribution to the uncertainty in A comes from the first derivative term, that is,

A ≅ A(r₀) + dA/dr(r₀) δr = A₀ + δA.

In this case, δA ≅ 2Cr δr. The relative uncertainty is given as δA/A₀ ≅ 2(δr/r₀).

Repeating this exercise with V = Dr³ for constant D, we get :

δV ≅ 3Dr² δr . The relative uncertainty is δV/V₀ ≅ 3(δr/r₀).

Hope this helps