To solve this problem, let's break it down step by step.

Given Information:

• At noon:

  • Ship A is 130 km west of Ship B.
  • Ship A moves east at 30 km/h.
  • Ship B moves north at 20 km/h.

• Goal: Find how fast the distance between the ships is changing at 4:00 p.m.

Steps to Solution:

1. Calculate Positions After 4 Hours:

   • Since 4 hours have passed (from noon to 4:00 p.m.):
     • Ship A: $4 \times 30 = 120$ km east.
     • Ship B: $4 \times 20 = 80$ km north.

2. Determine Distance Between Ships at 4:00 p.m.:

   • Initially, Ship A was 130 km west of Ship B. After 4 hours, Ship A has moved 120 km towards Ship B's initial position, so the horizontal distance between them is: $x = 130 - 120 = 10 \text{ km}$
   • The vertical distance (northward) between them is 80 km, as Ship B has traveled north.
   • Thus, the distance $D$ between the ships at 4:00 p.m. can be calculated using the Pythagorean theorem: $D = \sqrt{x^2 + y^2} = \sqrt{10^2 + 80^2} = \sqrt{100 + 6400} = \sqrt{6500} \approx 80.623 \text{ km}$

3. Differentiate to Find Rate of Change of Distance:

   • Let $x(t)$ and $y(t)$ represent the horizontal and vertical distances between the ships as functions of time $t$.
   • At 4:00 p.m., the rates of change are:
     • $\frac{dx}{dt} = -30$ km/h (since Ship A is moving towards Ship B, reducing $x$).
     • $\frac{dy}{dt} = 20$ km/h (as Ship B moves north).
   • Using the formula for the rate of change of the distance $D$: $\frac{dD}{dt} = \frac{x \cdot \frac{dx}{dt} + y \cdot \frac{dy}{dt}}{D}$
   • Substitute the values at 4:00 p.m.:
     • $x = 10$ km, $y = 80$ km, $D = 80.623$ km
     • $\frac{dx}{dt} = -30$ km/h, $\frac{dy}{dt} = 20$ km/h $\frac{dD}{dt} = \frac{(10)(-30) + (80)(20)}{80.623}$ $= \frac{-300 + 1600}{80.623} = \frac{1300}{80.623} \approx 16.125 \text{ km/h}$

Answer:

The rate at which the distance between the ships is changing at 4:00 p.m. is approximately 16.125 km/h.

Would you like more details on any part of this solution or have any questions?

Related Questions

1. What if Ship A was moving at a different speed? How would that affect the rate of change of distance?
2. How would the solution change if Ship B were moving south instead of north?
3. What if the problem asked for the distance between the ships at a different time?
4. Can we calculate when the distance between the ships was smallest?
5. How would we solve this problem if the ships had different initial positions?

Tip:

When dealing with related rates, always identify which variables change with respect to time and set up an equation involving their rates of change.