The $ABCD$ square-shaped field, measuring 19.36 hectares, has a point $E$ on the $AD$ side, 110 meters from $D$, to which a straight path leads from $B$. Antal and Béla are having a race. Antal starts from point $A$, and Béla from point $B$, both starting at the same time and running at a constant speed towards the common goal, point $D$. Béla reaches $D$ via the segments $BE + ED$. When Béla reaches $E$, Antal has a 30 m lead. Who wins the race and by how many meters?
The area of the square is $19.36 \mathrm{ha} = 193600 \mathrm{~m}^{2}$, so its side $AB = \sqrt{193600} = 400 \mathrm{~m}$. Therefore, $AE = AD - DE = 440 - 110 = 330 \mathrm{~m}$, and thus, according to the Pythagorean theorem, $BE = \sqrt{440^{2} + 330^{2}} = \sqrt{4^{2} \cdot 110^{2} + 3^{2} \cdot 110^{2}} = \sqrt{25 \cdot 110^{2}} = 5 \cdot 110 = 550 \mathrm{~m}$.

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According to the problem, while Béla ran 550 meters, Antal covered $330 + 30 = 360$ meters. Béla still has $110 \mathrm{~m}$ left, which is $1/5$ of his current distance. While Béla completes this and reaches the finish line, Antal will also cover $1/5$ of his current distance, which is $\frac{360}{5} = 72$ meters. But $30 + 72$ is only 102, so Béla wins the race by 8 meters.