Two ships are `10sqrt(2)` km apart on a line running south to north. The one further north is moving west with a speed of 25 km/h, while the other towards north with a speed of 25 km/h.

Options

A. Their distance of closest approach is `5sqrt(2)` km.
B. Their distance of closest approach is 10 km
C. Time required to reach the closest distance is 17 min
D. Time required to reach the closest distance is 29 min.

Explanation

To solve the problem, we need to determine the distance of closest approach between the two ships and the time taken to reach that distance. Let's break it down step by step.

### Step 1: Understand the Initial Setup
- We have two ships, A and B.
- Ship A is moving west at a speed of 25 km/h.
- Ship B is moving north at a speed of 25 km/h.
- The initial distance between the two ships is \(10\sqrt{2}\) km.

### Step 2: Visualize the Motion
- Ship A is moving horizontally (west), while Ship B is moving vertically (north).
- The initial distance between them forms a right triangle where:
- One leg is the distance Ship A is moving west.
- The other leg is the distance Ship B is moving north.

### Step 3: Relative Velocity
To find the distance of closest approach, we need to calculate the relative velocity of Ship B with respect to Ship A.

- The speed of Ship A (moving west) is \(V_A = 25\) km/h.
- The speed of Ship B (moving north) is \(V_B = 25\) km/h.

Since the two velocities are perpendicular to each other, we can use the Pythagorean theorem to find the relative speed \(V_{BA}\) of Ship B with respect to Ship A:

\[ V_{BA} = \sqrt{V_A^2 + V_B^2} = \sqrt{(25)^2 + (25)^2} = \sqrt{625 + 625} = \sqrt{1250} = 25\sqrt{2} \text{ km/h} \]

### Step 4: Finding the Distance of Closest Approach
The closest distance occurs when the line connecting the two ships is perpendicular to the direction of motion of either ship.

Using trigonometry, we can find the distance of closest approach \(D\) using the sine of the angle formed by the paths of the ships. The angle \(\theta\) can be calculated as:

\[ \tan \theta = \frac{V_B}{V_A} = \frac{25}{25} = 1 \Rightarrow \theta = 45^\circ \]

Using the sine function:

\[ D = \sin(45^\circ) \times PQ \]

Where \(PQ\) is the initial distance between the ships:

\[ D = \frac{1}{\sqrt{2}} \times 10\sqrt{2} = 10 \text{ km} \]

### Step 5: Calculate the Time to Reach Closest Distance
To find the time \(T\) taken to reach the closest distance, we can use the formula:

\[ T = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{PQ \cdot \cos(45^\circ)}{V_{BA}} \]

Substituting the values:

\[ T = \frac{10\sqrt{2} \cdot \frac{1}{\sqrt{2}}}{25\sqrt{2}} = \frac{10}{25\sqrt{2}} = \frac{10}{25\sqrt{2}} \cdot \frac{60}{60} = \frac{600}{25\sqrt{2}} = \frac{24}{\sqrt{2}} \text{ hours} \]

Converting to minutes:

\[ T = \frac{24 \times 60}{\sqrt{2}} \approx 17 \text{ minutes} \]

### Summary of Results
1. The distance of closest approach is \(10\) km.
2. The time required to reach the closest distance is approximately \(17\) minutes.