imc

If point B is due east of point A and point C is due north of point B, $\angle CBA$ is a right angle. And if $\angle BAC=45^{\circ }$, $△CBA$ is a 45-45-90 triangle. Thus, the lengths of sides $CB$, $BA$, and $AC$ are in the ratio $1:1:\sqrt{2}$, and $CB$ is $10\sqrt{2}÷\sqrt{2}=10$. $△DBA$ is clearly a right triangle with $C$ on the side $DB$. $DC$ is 20, so $DB=DC+CB=20+10=30$. By the Pythagorean Theorem, $DA=\sqrt{D{B}^2+B{A}^2}=\sqrt{30^2+10^2}=\sqrt{1000}$. $31^2=961$, and $32^2=1024$. Thus, $\sqrt{1000}$ must be between $31$ and $32$. The answer is $\boxed{31\text{and}32}$.