To determine between which two consecutive positive integers the positive square root of 262 lies, we need to find the approximate value of $\sqrt{262}$.

First, let's find the squares of the given integer pairs:

• $14^2 = 196$
• $15^2 = 225$
• $16^2 = 256$
• $17^2 = 289$
• $18^2 = 324$
• $19^2 = 361$

Now, let's analyze:

• $16^2 = 256$ is less than 262.
• $17^2 = 289$ is greater than 262.

Since $256 < 262 < 289$, the square root of 262 lies between 16 and 17.

Answer: The positive square root of 262 lies between the consecutive positive integers 16 and 17.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can you estimate the square root of a non-perfect square?
2. What is the method to calculate square roots manually?
3. Can you solve for the square root of 270 using the same approach?
4. How would you find the square root of 262 using a calculator?
5. What is the significance of square roots in real-world applications?

Tip: When estimating square roots, it’s useful to know the squares of numbers around the target value to quickly identify between which integers the root lies.