Which option states the position of ( \sqrt{15} ) on the number line most correctly?

Correct answer: Between (3.8) and (3.9)

The direct answer is option A: between 3.8 and 3.9. To locate a positive square root, compare squares because squaring preserves order for non-negative numbers. We have \(3.8^2=14.44\) and \(3.9^2=15.21\). Since \(14.44<15<15.21\), taking square roots gives \(3.8<\sqrt{15}<3.9\). Therefore option A is correct. Option B, between 3.5 and 3.6, is wrong because their squares are 12.25 and 12.96, both below 15. Option C, between 4.1 and 4.2, is wrong because their squares, 16.81 and 17.64, are above 15. Option D, between 2.9 and 3.0, is wrong because their squares are 8.41 and 9, far below 15. The nearby whole squares also help: \(3^2=9<15<16=4^2\), so the answer must lie between 3 and 4. Exam cue: square both proposed decimal boundaries.