Question

At noon, ship $A$ is $100$ km west of ship $B$. Ship $A$ is sailing east at $35$ km/hr and ship $B$ is sailing north at $25$ km/hr. How fast is the distance between the ships changing at $4.00$ p.m.?

Answer 1

Let $x$ and $y$ be the initial position of the ships $A$ and $B$ respectively.

Let $x$ be the distance between $A$ and $X,y$ be the distance between $B$ and $X,z$ be the distance between $A$ and $B$.
Then, $z^2=x^2+y^2$
$2z\frac{dz}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}$ ......... (1)
$z\frac{dz}{dt}=x\frac{dx}{dt}+y\frac{dy}{dt}$
Given, $\frac{dx}{dt}=35$ km/hr; $\frac{dy}{dt}=25$ km/hr
When $t=4$ hrs, $x=40;y=100$ and
$z=\sqrt{x^2+y^2}=\sqrt{(40{)}^2+(100{)}^2}$
$=\sqrt{1600+10000}=\sqrt{11600}=20\sqrt{29}$
$\left(1\right)\Rightarrow 20\sqrt{29}\frac{dz}{dt}=40\times \left(35\right)+100\times \left(25\right)$
$\frac{dz}{dt}=\frac{3900}{20\sqrt{29}}=\frac{195}{\sqrt{29}}$ km/hr