Map scale reduction

Cyril marked a square plot of land on a map with a scale of 1 ∶ 50,000 and calculated that its side corresponds to 1 km. He reduced the map on the copier so that the marked square had an area smaller by 1.44 cm² than on the original map. What was the scale of the map after scaling?

Final Answer:

M₂ =  1:62500

Step-by-step explanation:

$$\begin{gathered} M_1=50000 \\ {a}_1=1\text{km}\to \text{m}=1\cdot 1000\text{m}=1000\text{m} \\ {b}_1=a_1/M_1=1000/50000=\frac{1}{50}=0.02\text{m} \\ {c}_1=b_1\to \text{cm}=b_1\cdot 100\text{cm}=0.02\cdot 100\text{cm}=2\text{cm} \\ {S}_1=c_1^2=2^2=4{\text{cm}}^2 \\ {S}_2=S_1-1.44=4-1.44=\frac{64}{25}=2\frac{14}{25}=2.56{\text{cm}}^2 \\ {c}_2=\sqrt{S_2}=\sqrt{2.56}=\frac{8}{5}=1\frac{3}{5}=1.6\text{cm} \\ {b}_2=c_2\to \text{m}=c_2:100\text{m}=1.6:100\text{m}=0.016\text{m} \\ {a}_2=a_1=1000\text{km} \\ {M}_2=b_2/a_2=0.016/1000\approx 2.0\times 10^{-5}=1:62500 \end{gathered}$$

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