Calculating Area on a Map Using Scale

The question asks us to find the area represented on a map when we are given the map's scale and the actual area on the ground. The scale tells us how much real-world distance is represented by a certain distance on the map.

Given information:

• Map scale: 1 cm on the map represents 20 km in the real world.
• Real-world area of the country: 60,000 km².

Step 1: Understand the Linear Scale

The linear scale is given as 1 cm on the map equals 20 km in reality. We can write this as:

\(1 \text{ cm (map)} = 20 \text{ km (real)}\)

Step 2: Convert Linear Scale to Area Scale

To find out what area on the map represents a certain area in the real world, we need to convert the linear scale to an area scale. Area is measured in square units. If 1 cm on the map represents 20 km in reality, then 1 cm² on the map represents the area of a square with sides of 1 cm on the map, which correspond to sides of 20 km in reality.

So, the area represented by 1 cm² on the map is:

\(1 \text{ cm}^2 \text{ (map)} = (20 \text{ km} \times 20 \text{ km}) \text{ (real)}\)

\(1 \text{ cm}^2 \text{ (map)} = 400 \text{ km}^2 \text{ (real)}\)

This means that every square centimeter on the map represents 400 square kilometers of the actual country area.

Step 3: Calculate the Map Area for the Given Real Area

We know the total real-world area is 60,000 km². We want to find out how many units of 400 km² are in 60,000 km², because each 400 km² corresponds to 1 cm² on the map.

Let \(A_{\text{map}}\) be the area on the map in cm² that represents the real area. The relationship is:

\(A_{\text{map}} \times (\text{Area represented by 1 cm}^2) = \text{Real Area}\)

\(A_{\text{map}} \times 400 \text{ km}^2/\text{cm}^2 = 60,000 \text{ km}^2\)

To find \(A_{\text{map}}\), we divide the total real area by the area represented by 1 cm²:

\(A_{\text{map}} = \frac{\text{Real Area}}{\text{Area represented by 1 cm}^2}\)

\(A_{\text{map}} = \frac{60,000 \text{ km}^2}{400 \text{ km}^2/\text{cm}^2}\)

Now, perform the calculation:

\(A_{\text{map}} = \frac{60000}{400} \text{ cm}^2\)

\(A_{\text{map}} = \frac{600}{4} \text{ cm}^2\)

\(A_{\text{map}} = 150 \text{ cm}^2\)

Conclusion

An area of 60,000 km² in the real world will be represented by an area of 150 cm² on the map with a scale of 1 cm for 20 km.