Answer
The ships are moving apart at a rate of $55.4~km/h$
Work Step by Step
Let $x$ be the east-west distance between the two ships.
Then $x = 100~km$
Let $y$ be the north-south distance between the two ships. We can find $y$ at 4:00 pm:
$y = (35~km/h)(4~h)+ (25~km/h)(4~h)$
$y = 240~km$
Let $z$ be the distance between the two ships. We can find $z$ at 4:00 pm:
$z^2 = x^2+y^2$
$z = \sqrt{x^2+y^2}$
$z = \sqrt{(100~km)^2+(240~km)^2}$
$z = 260~km$
We can differentiate both sides of the equation with respect to $t$:
$z^2 = x^2+y^2$
$2z~\frac{dz}{dt} = 2x~\frac{dx}{dt} + 2y~\frac{dy}{dt}$
$\frac{dz}{dt} = \frac{1}{z}~(x~\frac{dx}{dt} + y~\frac{dy}{dt})$
$\frac{dz}{dt} = \frac{1}{260}~[(100)(0) + (240)(25+35)]$
$\frac{dz}{dt} = 55.4~km/h$
The ships are moving apart at a rate of $55.4~km/h$
