Two cyclists A and B run their cycles at average speed of 16 km/hr and 12 km/hr respectively. If A runs in North direction and B in East direction beginning from the same origin at the same time, what will be the minimum distance between the two after half an hour?

Options

A. 14 km
B. `2sqrt(14)` km
C. `sqrt(10)` km
D. 10 km

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Determine the distance traveled by each cyclist
Cyclist A travels north at a speed of 16 km/hr, and Cyclist B travels east at a speed of 12 km/hr. Since they both start at the same time and travel for half an hour (0.5 hours), we can calculate the distance each cyclist travels.

- **Distance traveled by A**:
\[ \text{Distance}_A = \text{Speed}_A \times \text{Time} = 16 \, \text{km/hr} \times 0.5 \, \text{hr} = 8 \, \text{km} \]

- **Distance traveled by B**:
\[ \text{Distance}_B = \text{Speed}_B \times \text{Time} = 12 \, \text{km/hr} \times 0.5 \, \text{hr} = 6 \, \text{km} \]

### Step 2: Visualize the positions of the cyclists
After half an hour:
- Cyclist A is 8 km north of the origin.
- Cyclist B is 6 km east of the origin.

### Step 3: Form a right triangle
The positions of the cyclists can be represented as points on a coordinate plane:
- Cyclist A's position: (0, 8)
- Cyclist B's position: (6, 0)

The distance between the two cyclists forms the hypotenuse of a right triangle, where:
- One side (vertical) is the distance traveled by A (8 km).
- The other side (horizontal) is the distance traveled by B (6 km).

### Step 4: Use the Pythagorean theorem to find the distance
To find the minimum distance between the two cyclists, we apply the Pythagorean theorem:
\[ \text{Distance}^2 = (\text{Distance}_A)^2 + (\text{Distance}_B)^2 \]
\[ \text{Distance}^2 = (8 \, \text{km})^2 + (6 \, \text{km})^2 \]
\[ \text{Distance}^2 = 64 + 36 = 100 \]
\[ \text{Distance} = \sqrt{100} = 10 \, \text{km} \]

### Final Answer
The minimum distance between the two cyclists after half an hour is **10 km**.

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