Differentiate with respect to time t the equation that relates everything.

\(\displaystyle{x}^{{{2}}}+{1}^{{{2}}}={y}^{{{2}}}\)

\(\displaystyle{2}{x}{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}+{0}={2}{y}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}\)

\(\displaystyle{x}{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}\)

\(\displaystyle{\frac{{{x}}}{{{y}}}}{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}\)

We are given that a \(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={500}{\frac{{{m}{i}}}{{{h}}}}\). We also need to find x at the moment \(\displaystyle{y}={2}{m}{i}\)

\(\displaystyle{x}^{{{2}}}+{1}^{{{2}}}={2}^{{{2}}}\)

\(\displaystyle{x}^{{2}}={4}-{1}={3}\)

\(\displaystyle{x}=\sqrt{{{3}}}\) (ignore negative root)

Plug the values into the differentiated equation.

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{x}}}{{{y}}}}{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}\)

\(\displaystyle{\frac{{\sqrt{{{3}}}}}{{{2}}}}\cdot{500}\)

\(\displaystyle{250}\sqrt{{{3}}}{\frac{{{m}{i}}}{{{h}}}}\)