#### Answer

$k=4$, $y=\frac{4}{x}$

#### Work Step by Step

If y varies inversely as x, we know that the variables x and y can be related by the function $y=\frac{k}{x}$, where k is the constant of proportionality.
If $y=\frac{1}{10}$ when $x=40$, then we know that $\frac{1}{10}=\frac{k}{40}$. Multiply both sides by 40.
$k=\frac{40}{10}=4$
Therefore, the constant of variation is $k=4$ and the inverse variation equation is $y=\frac{4}{x}$.