A water dam is drawn using a scale of 1 : 2 500. Determine the scale used in drawing the trench is 10 cm by 1 cm, cm long and 8 cm wide, find total area in hectares that is required for the construction of the dam.

1. Step 1: Identify the given information. The height of the tree on paper is $10 \text{ cm}$. The scale used is $1:500$. This means $1 \text{ cm}$ on paper represents $500 \text{ cm}$ in actual height.

Step 2: Calculate the actual height of the tree. Actual height = Diagram height $\times$ Scale factor $Actual height = 10 cm \times 500 = 5000 cm$

Step 3: Convert the actual height from centimeters to meters. Since $1 m = 100 \text{ cm}$: $Actual height = \frac{5000 cm}{100 cm/m} = 50 m$ The actual height of the tree is $\boxed{50 m}$.

2. Step 1: Identify the given dimensions for the trench. Actual length = $50 \text{ m}$ Actual depth = $2.5 \text{ m}$ Diagram length = $10 \text{ cm}$ Diagram depth = $0.5 \text{ cm}$

Step 2: Convert actual dimensions to centimeters to match the diagram units. Actual length in cm = $50 m \times 100 cm/m = 5000 \text{ cm}$ Actual depth in cm = $2.5 m \times 100 cm/m = 250 \text{ cm}$

Step 3: Determine the scale using the length. Scale = Diagram length : Actual length $Scale = 10 cm : 5000 cm$ To express this in the form $1:N$, divide both sides by $10$: $Scale = \frac{10}{10} : \frac{5000}{10} = 1 : 500$ (We can verify this with depth: $0.5 cm : 250 \text{ cm}$. Divide both by $0.5$: $1 : \frac{250}{0.5} = 1 : 500$). The scale used is $\boxed{1:500}$.

3. Step 1: Identify the given information for the water dam. Diagram length = $10 \text{ cm}$ Diagram width = $8 \text{ cm}$ Scale used = $1:2500$

Step 2: Calculate the actual dimensions of the dam in centimeters. Actual length = Diagram length $\times$ Scale factor $Actual length = 10 cm \times 2500 = 25000 cm$ Actual width = Diagram width $\times$ Scale factor $Actual width = 8 cm \times 2500 = 20000 cm$

Step 3: Convert the actual dimensions from centimeters to meters. Actual length in m = $\frac{25000 cm}{100 cm/m} = 250 \text{ m}$ Actual width in m = $\frac{20000 cm}{100 cm/m} = 200 \text{ m}$

Step 4: Calculate the actual area of the dam in square meters. Area = Actual length $\times$ Actual width $Area = 250 m \times 200 m = 50000 m^2$

Step 5: Convert the area from square meters to hectares. Since $1 hectare = 10000 m^2$: $Area in hectares = \frac{50000 m^2}{10000 m^2/hectare} = 5 hectares$ The total area required for the construction of the dam is $\boxed{5 hectares}$.