Question

on a scale map of a city, an area of one square kilometer is represented by a square with an area of 4 square centimeters. if two buildings in the city are 3.5 kilometers apart, what is the distance between the buildings on the map, in centimeters? competency c08 7 28 21 14

Explanation:

Step1: Find the scale for linear distance from area scale

The area of 1 square kilometer is represented by 4 square centimeters. For a square, area of a square is side length squared. Let the side length (linear scale) for 1 km be \( s \) cm. So \( s^2 = 4 \), then \( s=\sqrt{4} = 2 \) cm. This means 1 kilometer in real is represented by 2 centimeters on the map.

Step2: Calculate the map distance for 3.5 km

If 1 km is 2 cm, then for 3.5 km, the distance on the map is \( 3.5\times2 = 7 \) cm. Wait, no, wait. Wait, maybe I made a mistake. Wait, the area scale: 1 km² is 4 cm². So the linear scale factor: if real area \( A_{real}=1\ km^2=(1\ km\times1\ km) \), map area \( A_{map}=4\ cm^2=(2\ cm\times2\ cm) \). So the linear scale is 2 cm represents 1 km? Wait, no, 1 km in real is 2 cm in map? Wait, no, the side of the square representing 1 km² is 1 km (real) and 2 cm (map), because area is side squared. So 1 km (real) corresponds to 2 cm (map). Then 3.5 km real would be 3.5 2 cm? Wait, but 3.52=7? But let's check again. Wait, maybe the linear scale is different. Wait, area scale is (linear scale)². So if \( A_{map} = k^2\times A_{real} \), where \( k \) is the linear scale factor (map units per real unit). So \( 4\ cm^2 = k^2\times1\ km^2 \), so \( k = \sqrt{\frac{4\ cm^2}{1\ km^2}} = \frac{2\ cm}{1\ km} \). So linear scale is 2 cm per km. Then distance between buildings is 3.5 km, so map distance is 3.5 km 2 cm/km = 7 cm. Wait, but the options have 7 as an option. But wait, maybe I messed up. Wait, 3.5 km is the distance, so linear distance. So linear scale: 1 km (real) is 2 cm (map). So 3.5 km 2 cm/km = 7 cm. So the answer is 7.

Answer:

7