Answer:
see attached
Explanation:
Your calculator will tell you that √19 ≈ 4.36. The marks on the given number line are spaced by 1/3 unit, so the mark immediately to the right of 4 will be 4 1/3, approximately 4.33. You need to place your "click" very slightly to the right of the mark at 4 1/3.
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As a first approximation, a square root can be estimated by the number's difference from the nearest (lower) square.
We know that 19 = 16 +3 = 4² +3. The root can be estimated to be ...
Using this approximation, the estimated root would be on the first mark to the right of 4 on your number line.
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The second attachment shows a number line with the point plotted by a graphing calculator.
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Additional comment
The above square root estimate is based on a linear interpolation between the roots of 16 = 4² and 25 = 5². That is usually described as ...
√n ≈ 4 +(n -16)/(25 -16) . . . . for 16 ≤ n < 25
The denominator difference (25 -16) is 9, which is 1 more than double the integer part of the root (=2·4 +1). This will always be the case for this sort of interpolation.