A map

A map with a scale of 1: 5,000 shows a rectangular field with an area of 18 ha. The length of the field is three times its width. The area of the field on the map is 72 cm square. What is the actual length and width of the field?

Correct result:

a =  734.8469 m
b =  244.949 m

Solution:

$$\begin{gathered} M=5000 \\ S=18\text{ ha}\to {\text{m}}^2=18\cdot 10000{\text{m}}^2=180000{\text{m}}^2 \\ a=3b \\ S=ab \\ S=a(a\mathrm{/}3) \\ S=a^2\mathrm{/}3 \\ a=\sqrt{3\cdot S}=\sqrt{3\cdot 180000}=300\sqrt{6}=734.8469\text{ m} \end{gathered}$$

$$\begin{gathered} S_2=S\to {\text{cm}}^2=S\cdot 10000{\text{cm}}^2=180000\cdot 10000{\text{cm}}^2=1800000000{\text{cm}}^2 \\ {S}_3=S_2\mathrm{/}{M}^2=1800000000\mathrm{/}5000^2=72{\text{cm}}^2 \\ b=a\mathrm{/}3=734.8469\mathrm{/}3=100\sqrt{6}=244.949\text{ m} \end{gathered}$$

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