Question

At what position on the number line is the red dot located?

Answer 1

√42

Explanation:

The red dot is around the 6 point but less than the 7 point. Therefore, the square root of 42, being 6.4, would be the most accurate pick here.

Hope this helps!

Answer 2

The red dot is at location x

x is between 6 and 7

`6 < \text{x} < 7\\\\6^2 < \text{x}^2 < 7^2\\\\36 < \text{x}^2 < 49\\\\`

This means,

`36 < \text{x}^2 < 49\\\\√(36) < \sqrt{\text{x}^2} < √(49)\\\\√(36) < \text{x} < √(49)\\\\√(36) < \sqrt{\text{y}} < √(49)`

In other words, the stuff under the square root is between 36 and 49, excluding both endpoints.

That rules out choices A and D.

As for the remaining choices B and C, they both appear equally valid since the radicands are between 36 and 49.

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But notice how the red dot appears to be at or very close to 6.5

Furthermore, squaring that value gets to (6.5)^2 = (6.5)*(6.5) = 42.25

In other words,
`√(42.25) = 6.5`

So it appears that the red dot is at location
`√(42)`