To solve the problem, we can use the principles of related rates in calculus.

1. Position of the Ships at Noon:

   • Ship A is 170 km west of Ship B.
   • Let the position of Ship B be the origin (0, 0).
   • Position of Ship A: $(-170, 0)$.

2. Speeds:

   • Ship A is sailing east at 35 km/h, so its x-coordinate increases.
   • Ship B is sailing north at 20 km/h, so its y-coordinate increases.

3. Positions at 4:00 PM:

   • Time from noon to 4 PM is 4 hours.
   • Position of Ship A at 4 PM: $x_A = -170 + 35 \times 4 = -170 + 140 = -30 \text{ km}$
   • Position of Ship B at 4 PM: $y_B = 0 + 20 \times 4 = 80 \text{ km}$

4. Distance Between the Ships: Let $d$ be the distance between the two ships: $d = \sqrt{(x_A - x_B)^2 + (y_A - y_B)^2}$ In our case: $d = \sqrt{(-30 - 0)^2 + (0 - 80)^2} = \sqrt{(-30)^2 + (-80)^2} = \sqrt{900 + 6400} = \sqrt{7300} \approx 85.44 \text{ km}$

5. Using Related Rates: Let $x = x_A + 170$ (position of Ship A from Ship B) and $y = y_B$. Then: $x = -30 + 170 = 140 \text{ km} \quad (since \, x_A \, is \, west \, of \, B)$ The rate of change of $x$ with respect to time $t$ is $\frac{dx}{dt} = 35 \, \text{km/h}$ (east) and the rate of change of $y$ is $\frac{dy}{dt} = 20 \, \text{km/h}$ (north).

6. Differentiating the Distance Formula: Using the formula for distance: $d^2 = x^2 + y^2$ Differentiating with respect to time $t$: $2d \frac{dd}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$ Rearranging gives: $\frac{dd}{dt} = \frac{x \frac{dx}{dt} + y \frac{dy}{dt}}{d}$

7. Substituting Values: At 4 PM:

   • $x = 140 \text{ km}$
   • $y = 80 \text{ km}$
   • $d \approx 85.44 \text{ km}$
   • $\frac{dx}{dt} = 35 \, \text{km/h}$
   • $\frac{dy}{dt} = 20 \, \text{km/h}$

   Now calculate: $\frac{dd}{dt} = \frac{(140)(35) + (80)(20)}{85.44}$ $= \frac{4900 + 1600}{85.44} = \frac{6500}{85.44} \approx 76.06 \text{ km/h}$

Therefore, the distance between the ships is increasing at approximately 76.06 km/h at 4:00 PM.

Would you like more details on any part of this problem, or do you have other questions? Here are some related questions you might find interesting:

1. How would the distance change if Ship A was sailing west instead?
2. What would be the distance between the ships at 5 PM?
3. How would the problem change if Ship B was sailing south instead of north?
4. What is the total distance covered by each ship by 4 PM?
5. How can we visualize the movement of the ships over time?

Tip: Always diagram the situation for related rates problems; it can clarify how the distances and directions relate to each other.