Ship A and Ship B

Question

At noon, ship A is 90 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 5 km/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.)

Answer 1

Sketch of problem:

B’

^

| 5t

A x——————————x B

|

| 25 t

|

v

A’

We want to figure out the rate of change for distance x between A’ and B’ after t = 4.

x at t(0) = 90 km between A and B

x at t(1) = sqrt( 90² + 30²⁾ = sqrt(9000) is approx 95 km. (Use Pythagoras for distance)

in general, x = sqrt(90² + (25t + 5t)² ) = sqrt(1800 + 900t²)

rate of change dx/dt = 1/2 (1800t) / sqrt(1800 + 900t²)) (take derivative based on t)_

after t=4 hours: dx/dt = 1/2 (1800(4) / sqrt (1800 + 900(4²)) = 24 km/h.