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.

$$\begin{array}{c}x = \left( 2 \right)\left( {15} \right) = 30\hspace{1.0in}y = \left( 7 \right)\left( {15} \right) = 105\\ z = \sqrt {{{30}^2} + {{105}^2}} = \sqrt {11925} = 15\sqrt {53} = 109.2016\end{array}$$

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

$$z' = \frac{{\left( {30} \right)\left( 2 \right) + \left( {105} \right)\left( 7 \right)}}{{109.2016}} = \require{bbox} \bbox[2pt,border:1px solid black]{{7.2801}}$$