To find which whole numbers have squares between 50 and 200, we need to identify the integers $n$ such that:

$50 \leq n^2 \leq 200$

Step 1: Solve for lower bound ($n^2 \geq 50$)

We need to solve $n^2 \geq 50$. Taking the square root of both sides:

$n \geq \sqrt{50} \approx 7.07$

Since $n$ must be a whole number, the smallest possible value of $n$ is 8.

Step 2: Solve for upper bound ($n^2 \leq 200$)

Now, solve $n^2 \leq 200$. Taking the square root of both sides:

$n \leq \sqrt{200} \approx 14.14$

Since $n$ must be a whole number, the largest possible value of $n$ is 14.

Step 3: Identify the integers between the bounds

We have found that $n$ must be a whole number between 8 and 14 (inclusive). Therefore, the possible values of $n$ are:

$n = 8, 9, 10, 11, 12, 13, 14$

Step 4: Check the squares of these numbers

Now let's check the squares of these numbers to ensure they fall between 50 and 200:

$8^2 = 64, \quad 9^2 = 81, \quad 10^2 = 100$ $11^2 = 121, \quad 12^2 = 144, \quad 13^2 = 169, \quad 14^2 = 196$

All these squares lie between 50 and 200.

Conclusion

The whole numbers whose squares are between 50 and 200 are:

$8, 9, 10, 11, 12, 13, 14$

Justification

We solved the inequality $50 \leq n^2 \leq 200$ by taking square roots and finding that $n$ must lie between approximately 7.07 and 14.14. This led to the whole numbers from 8 to 14. We then verified their squares, which all fall within the given range.

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Do you have any questions or would you like further details?

Related Questions:

1. What are the whole numbers whose squares are less than 50?
2. Which whole numbers have squares between 200 and 500?
3. How would you find numbers whose cubes fall within a specific range?
4. What is the difference between squares of consecutive whole numbers?
5. How can you generalize this process for any given range of squares?

Tip:

When dealing with inequalities involving squares, always consider taking square roots to narrow down the range of values.