1. **State the problem:** We have two maps showing the same forest. The first map has a scale of 1 : 20 000 and the forest area is 50 cm². The second map shows the forest area as 8 cm². We need to find the scale of the second map. 2. **Recall the relationship between scale and area:** The scale factor between two maps affects lengths linearly, but areas scale by the square of the scale factor. If the scale factor for lengths is $k$, then the area scale factor is $k^2$. 3. **Set up the equation:** Let the scale of the second map be 1 : $x$. The length scale factor from the first to the second map is $\frac{1/ x}{1/ 20000} = \frac{20000}{x}$. 4. **Relate the areas:** The ratio of areas is $$\frac{8}{50} = \left(\frac{20000}{x}\right)^2$$ 5. **Solve for $x$:** $$\frac{8}{50} = \frac{64}{400} = \left(\frac{20000}{x}\right)^2$$ Take the square root of both sides: $$\sqrt{\frac{8}{50}} = \frac{20000}{x}$$ $$\frac{\sqrt{8}}{\sqrt{50}} = \frac{20000}{x}$$ Simplify the square roots: $$\frac{2\sqrt{2}}{5\sqrt{2} \cdot \sqrt{5}} = \frac{20000}{x}$$ Cancel $\sqrt{2}$: $$\frac{2}{5\sqrt{5}} = \frac{20000}{x}$$ Multiply both sides by $x$ and divide both sides by $\frac{2}{5\sqrt{5}}$: $$x = 20000 \times \frac{5\sqrt{5}}{2}$$ Calculate: $$x = 20000 \times \frac{5\sqrt{5}}{2} = 10000 \times 5 \sqrt{5} = 50000 \sqrt{5}$$ Approximate $\sqrt{5} \approx 2.236$: $$x \approx 50000 \times 2.236 = 111800$$ 6. **Final answer:** The scale of the second map is approximately 1 : 111800.