503. By direct measurement, it was found that the diameter of a circle is 6.4 cm, with the maximum error not exceeding $0.05 \mathrm{~cm}$. Find approximately the maximum error in the estimation of the area, calculated by the formula $S=\frac{1}{4} \pi x^{2}$ ( $x$ - diameter).
Solution. Obviously, the exact value of the maximum error is $\Delta S$, where $x$ varies from 6.4 to 6.45. The approximate value of the maximum error is the corresponding differential $d S$. We find

$$
d S=S^{\prime}(x) d x=\left.\frac{1}{2} \pi x d x\right|_{\substack{x=6.4 \\ d x=0.05}}=\frac{1}{2} \pi \cdot 6.4 \cdot 0.05=0.5024\left(\mathrm{~cm}^{2}\right)
$$

The relative error in the area estimation is

$$
\frac{d S}{S}=\frac{2 d x}{x}=\frac{2 \cdot 0.05}{6.4} \approx 0.015=1.5 \%
$$