Hopefully it’s clear that we’ll need the Pythagorean Theorem to solve this problem so here is that.

$${z^2} = {x^2} + {y^2}$$ Show Step 3

Finally, let’s differentiate this with respect to $t$ and we can even solve it for $z'$ so the actual solution will be quick and simple to find.

$$2z\,z' = 2x\,x' + 2y\,y'\hspace{0.5in} \Rightarrow \hspace{0.5in}z' = \frac{{x\,x' + y\,y'}}{z}$$ Show Step 4

To finish off this problem all we need to do is determine all three lengths of the triangle in the sketch above. We can find $x$ and $y$ using their speeds and time while we can find $z$ by reusing the Pythagorean Theorem. Note that the biker riding east will be riding for 20 seconds and the biker riding north will be riding for 25 seconds (this biker started 5 seconds earlier…).

$$\begin{array}{c}x = \left( 5 \right)\left( {20} \right) = 100\hspace{0.5in}\hspace{0.25in}y = \left( 8 \right)\left( {25} \right) = 200\\ z = \sqrt {{{100}^2} + {{200}^2}} = \sqrt {50000} = 100\sqrt 5 = 223.6068\end{array}$$

The rate of change of the distance between the two people is then,

$$z' = \frac{{\left( {100} \right)\left( 5 \right) + \left( {200} \right)\left( 8 \right)}}{{223.6068}} = \require{bbox} \bbox[2pt,border:1px solid black]{{9.3915}}$$