\section*{Problem 21 - V01121}

During the surveying (measurement and determination of coordinates) of a country road, a polygonal line is obtained, whose vertices have the following coordinates (measurements in m):

\[
A(0,00; 0,00), B(87,00; 54,40), C(153,60; 44,00), D(206,40; 25,00), E(303,50; 33,80), F(352,00; 0,00)
\]

a) Calculate the length of the country road!

b) The country road is \(5.5 \mathrm{~m}\) wide. It is to be paved. It is to be estimated approximately how many \(\mathrm{m}^{2}\) of road need to be paved!
}

a) According to the Pythagorean theorem, we have:

\[
\begin{aligned}
& |\overline{A B}|=\sqrt{(87-0)^{2}+(54.4-0)^{2}} \approx 102.61 \\
& |\overline{B C}|=\sqrt{(153.6-87)^{2}+(44-54.4)^{2}} \approx 67.41 \\
& |\overline{C D}|=\sqrt{(206.4-153.6)^{2}+(25-44)^{2}} \approx 56.11 \\
& |\overline{D E}|=\sqrt{(303.5-206.4)^{2}+(33.8-25)^{2}} \approx 97.50 \\
& |\overline{E F}|=\sqrt{(352-303.5)^{2}+(0-33.8)^{2}} \approx 59.12
\end{aligned}
\]

The length \(l\) of the road is \(l \approx 382.75 \mathrm{~m}\). b) For the area to be asphalted, we have

\[
A=5.5 \mathrm{~m} \cdot l=5.5 \mathrm{~m} \cdot 382.75 \mathrm{~m} \approx 2105.13 \mathrm{~m}^{2}
\]

and thus approximately 2105.13 square meters of road need to be asphalted.