jmc

Since $2^3=8$ and $3^3=27$, we know that $2<\sqrt[3]{10}<3$. Then we find that $5^3=125$ and $6^3=216$, so $5<\sqrt[3]{200}<6$. We have $\sqrt[3]{10}<3$ and $5<\sqrt[3]{200}$. The whole numbers between $\sqrt[3]{10}$ and $\sqrt[3]{200}$ are $3,4,5$, for a total of $\boxed{3}$ whole numbers.