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 cm². What is the actual length and width of the field?

Final Answer:

a =  734.8469 m
b =  244.949 m

Step-by-step explanation:

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

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

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