The question asks which of the following is a possible coordinate of the point shown on the number line between 2 and 5.

Here are the answer choices:

• A: $3\pi$
• B: $\sqrt{3}$
• C: $\sqrt{10}$
• D: $\sqrt[3]{10}$

Let’s evaluate each option to determine if it falls between 2 and 5:

1. Option A: $3\pi$

   • $\pi \approx 3.1416$, so $3\pi \approx 3 \times 3.1416 = 9.4248$, which is greater than 5. Therefore, this is not a possible coordinate.

2. Option B: $\sqrt{3}$

   • $\sqrt{3} \approx 1.732$, which is less than 2. Therefore, this is not a possible coordinate.

3. Option C: $\sqrt{10}$

   • $\sqrt{10} \approx 3.162$, which falls between 3 and 4. Therefore, this is a possible coordinate.

4. Option D: $\sqrt[3]{10}$

   • $\sqrt[3]{10} \approx 2.154$, which is slightly greater than 2 but less than 3. Therefore, this is not a possible coordinate because the point on the number line is closer to 4.

Thus, the correct answer is C: $\sqrt{10}$.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore further:

1. What is the significance of irrational numbers like $\sqrt{10}$ on a number line?
2. How do you estimate square roots to place them accurately on a number line?
3. What is the cube root of 10, and how is it approximated?
4. How does $\pi$ relate to irrational numbers, and why is $3\pi$ so large?
5. How can square and cube roots be simplified or estimated for calculations?

Tip: When solving problems involving square roots, it helps to memorize common square root values (like $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, etc.).