Find the perfect square numbers between

(i) 30 and 40

(ii) 50 and 60.

To find perfect square numbers within specific ranges, we first need to understand what a perfect square is. A perfect square is an integer that can be expressed as the square of another integer. For example, 1 (1x1), 4 (2x2), 9 (3x3), and so forth. Now, let’s look at the ranges you provided: between 30 and 40, and between 50 and 60.

Perfect Squares Between 30 and 40

To identify perfect squares in this range, we can start by determining the square roots of the endpoints:

• The square root of 30 is approximately 5.48.
• The square root of 40 is approximately 6.32.

This means we should consider the integers 6 (since 5.48 rounds up) and below. The perfect squares of these integers are:

• 6 x 6 = 36

Thus, the only perfect square number between 30 and 40 is 36.

Perfect Squares Between 50 and 60

Next, we’ll analyze the second range. Again, we’ll find the square roots:

• The square root of 50 is approximately 7.07.
• The square root of 60 is approximately 7.75.

In this case, we consider the integer 7. The perfect squares of these integers are:

• 7 x 7 = 49
• 8 x 8 = 64

However, since 49 is less than 50 and 64 exceeds 60, there are no perfect squares in this range. Therefore, there are no perfect square numbers between 50 and 60.

Summary

In summary, the only perfect square between 30 and 40 is 36, while there are no perfect squares between 50 and 60. This exercise illustrates how to systematically approach finding perfect squares by using square roots and checking the integers within the specified ranges.