\section*{Task 2 - V11032}

Günther wants to calculate the area of a regular hexagon in a workpiece during a school day in socialist production. His supervisor claims that the approximate formula \(F_{S}=\frac{7}{8} a^{2}\) can be used, where \(a\) is the distance between two parallel sides of the hexagon.

a) What is the exact area formula?

b) What percentage of the exact value is the error when using the approximate formula for \(a=50 \mathrm{~mm}\)?
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![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1172.jpg?height=315&width=302&top_left_y=2224&top_left_x=340)

a) The exact formula is:

\[
F_{S}=\left(s+\frac{1}{2} s\right) a=\frac{3}{2} a s
\]

According to the Pythagorean theorem:

\[
a=\sqrt{(2 s)^{2}-s^{2}}=\sqrt{3} s \quad \rightarrow \quad s=\frac{a}{\sqrt{3}}
\]

Therefore, the exact formula is:

\[
F_{S}=\frac{3}{2 \sqrt{3}} a^{2}=\frac{\sqrt{3}}{2} a^{2}
\]
b) The percentage error is always the same, regardless of whether \(a=50 \mathrm{~mm}\) or any other value. The relative error is

\[
1-\frac{\frac{7}{8}}{\frac{\sqrt{3}}{2}}=1-\frac{7}{4 \sqrt{3}}=1-\frac{7}{12} \sqrt{3}
\]

This is approximately \(-1 \%\). (The approximation yields a larger value than the exact formula, hence negative).