To start the serial production of high quality bearings, engineers must measure, as accurately as possible, the radius R of a small metallic sphere that is part of a prototype. To do so, they have three different alternatives: 1. Measure the diameter D and get the radius R as R = D/2. 2. Measure the surface S using indirect techniques and get the radius as R=sqrt[S/(4π)]. a) Obtain the expressions of absolute and relative errors for these three formulas of R. You can leave them in function of an approximate value R*. b) Which of the three alternatives is the most suitable from the numerical point of view?
In the case that you need the relative errors of D, S , you may take the constant value 10^(−3) and you can use the value π = 3.14159265 ± 0.5 · 10*(-8)
Does anybody knows how to solve it? i have already two different solutions, both of them make sense to me, so i need a second opinion. thanks :)
As you wrote, we have $$R=\frac D 2\qquad , \qquad R=\sqrt{\frac S {4\pi}}$$ Take logarithms and use differentiation to get $$\frac {dR}R=\frac {dD}D\qquad ,\qquad \frac {dR}R=\frac 12\frac {dS}S$$ Then, going to $\Delta$'s $$\frac {\Delta R}R=\frac {\Delta D}D\qquad ,\qquad \frac {\Delta R}R=\frac 12\frac {\Delta S}S$$ So, if $\frac {\Delta D}D=\frac {\Delta S}S$, measuring the surface is twice better.