How many whole numbers are between $\sqrt[3]{10}$ and $\sqrt[3]{200}$?
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.
3
Level 2
Algebra
Problem: How many whole numbers are between $\sqrt[3]{10}$ and $\sqrt[3]{200}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the whole numbers between $\sqrt[3]{10}$ and $\sqrt[3]{200}$, we first need to calculate the approximate values of these cube roots. 

Calculating $\sqrt[3]{10}$, we know that $2^3 = 8$ and $3^3 = 27$. Therefore, $\sqrt[3]{10}$ is between 2 and 3. A more precise estimate can be found by recognizing that $2.15^3 \approx 9.94$ and $2.2^3 \approx 10.648$, indicating that $\sqrt[3]{10} \approx 2.15$.

Next, we calculate $\sqrt[3]{200}$. We know that $5^3 = 125$ and $6^3 = 216$. Thus, $\sqrt[3]{200}$ is between 5 and 6. A more precise estimate shows that $5.8^3 \approx 195.112$ and $5.9^3 \approx 205.379$, indicating that $\sqrt[3]{200} \approx 5.8$.

Now, we need to identify the whole numbers that lie between these two values. The whole numbers greater than $\sqrt[3]{10} \approx 2.15$ are 3, 4, and 5. The whole numbers less than $\sqrt[3]{200} \approx 5.8$ are also 3, 4, and 5.

Thus, the whole numbers between $\sqrt[3]{10}$ and $\sqrt[3]{200}$ are 3, 4, and 5. Counting these gives us a total of 3 whole numbers. 

Therefore, the final answer is 3.<|END_OF_LATENT|> The final answer is $3$.