To determine how far apart the two cars are after two hours, let's break down the problem step by step:
1. Understanding the Initial Conditions:
- Car A travels south at a speed of 100 km/hr.
- Car B travels west at a speed of 50 km/hr.
- Both cars start moving at the same time from the same point.
2. Calculating the Distance Traveled by Each Car:
- Time traveled by both cars is 2 hours.
For Car A (traveling south):
[tex]\[
\text{Distance traveled by Car A} = \text{Speed of Car A} \times \text{Time}
\][/tex]
[tex]\[
\text{Distance traveled by Car A} = 100 \text{ km/hr} \times 2 \text{ hours} = 200 \text{ km}
\][/tex]
For Car B (traveling west):
[tex]\[
\text{Distance traveled by Car B} = \text{Speed of Car B} \times \text{Time}
\][/tex]
[tex]\[
\text{Distance traveled by Car B} = 50 \text{ km/hr} \times 2 \text{ hours} = 100 \text{ km}
\][/tex]
3. Using the Pythagorean Theorem to Find the Distance Apart:
Since the cars are traveling in perpendicular directions (south and west), the distance between them forms the hypotenuse of a right-angled triangle where the two legs of the triangle are the distances traveled by each car.
- Let [tex]\(d_s\)[/tex] be the distance traveled by Car A (south) = 200 km.
- Let [tex]\(d_w\)[/tex] be the distance traveled by Car B (west) = 100 km.
By the Pythagorean theorem:
[tex]\[
\text{Distance apart (d)} = \sqrt{(d_s)^2 + (d_w)^2}
\][/tex]
[tex]\[
\text{Distance apart (d)} = \sqrt{(200 \text{ km})^2 + (100 \text{ km})^2}
\][/tex]
[tex]\[
\text{Distance apart (d)} = \sqrt{40000 \text{ km}^2 + 10000 \text{ km}^2}
\][/tex]
[tex]\[
\text{Distance apart (d)} = \sqrt{50000 \text{ km}^2}
\][/tex]
[tex]\[
\text{Distance apart (d)} = 223.60679774997897 \text{ km}
\][/tex]
Therefore, after two hours, the two cars are approximately 223.61 km apart.
