Answer:
The distance between the ships is changing at the rate of 113.9km/hr
Explanation:
Given that A is 100 km west of B. and A is sailing south at 40 km/h,
The coordinate of A is (0,-40t)
The coordinate of B is (100,30t)
Where t represent time from initial point to final point
Calculating the distance between both ships
d(t) = √((100 - 0)² + (30t - (-40t))
d(t) = √((100)² + (30t + 40t))
d(t) = √(10000 + (70t)²)
d(t) = √(10000 + 4900t²)
Differentiate with respect to t
d'(t) = ½(1/√(10000 + 4900t²))*9800t
d'(t) = 4900t/(√(10000 + 4900t²))
Simplify to lowest term
d'(t) = 4900t/(√(100(100+49t²)))
d'(t) = 4900t/(10√(100+49t²))
d'(t) = 490t/√(100+49t²)
Substitute 4 for t in the above
d'(4) = 490*4/√(100+49*4²)
d'(4) = 1960/√296
d'(4) = 113.9226859689428
d'(4) = 113.9
The distance between the ships is changing at the rate of 113.9km/hr