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?

Result

a =  734.847 m
b =  244.949 m

Solution:

M=5000 S=18 ha→m2=18⋅ 10000 m2=180000 m2 a=3b  S=ab S=a(a/3) S=a2/3  a=3⋅ S=3⋅ 180000≐300 6≐734.8469≐734.847 mM=5000 \ \\ S=18 \ ha \rightarrow m^2=18 \cdot \ 10000 \ m^2=180000 \ m^2 \ \\ a=3b \ \\ \ \\ S=ab \ \\ S=a(a/3) \ \\ S=a^2/3 \ \\ \ \\ a=\sqrt{ 3 \cdot \ S }=\sqrt{ 3 \cdot \ 180000 } \doteq 300 \ \sqrt{ 6 } \doteq 734.8469 \doteq 734.847 \ \text{m}
S2=S→cm2=S⋅ 10000 cm2=180000⋅ 10000 cm2=1800000000 cm2 S3=S2/M2=1800000000/50002=72 cm2 b=a/3=734.8469/3=244.949 mS_{2}=S \rightarrow cm^2=S \cdot \ 10000 \ cm^2=180000 \cdot \ 10000 \ cm^2=1800000000 \ cm^2 \ \\ S_{3}=S_{2}/M^2=1800000000/5000^2=72 \ \text{cm}^2 \ \\ b=a/3=734.8469/3=244.949 \ \text{m}

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