= Solution
Converting all lengths to metres gives $d=3.7276\times10^{-5}\,\mathrm m$. <Logarithmic differentiation> gives
$$
\frac{\delta d}{d}=\frac{\delta F}{F}+3\frac{\delta L}{L}-\frac{\delta E}{E}-\frac{\delta w}{w}-3\frac{\delta t}{t}.
$$
If the quoted uncertainties are <independent> <standard uncertainties>, <monomial uncertainty propagation> gives
$$
\frac{\sigma_d}{d}=\sqrt{0.01^2+0.06^2+0.005^2+(0.1/9)^2+0.025^2}=0.0669,
\qquad
\boxed{d=(37.3\pm2.5)\,\mathrm{\mu m}.}
$$
If the quoted uncertainties instead mean bounds, the first-order conservative fractional error is $0.111$, giving \b[approximately $37.3\pm4.1\,\mathrm{\mu m}$]. The question does not specify their statistical meaning; these two error conventions should not be combined.
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