Bounds, error intervals, and multiplying measurements – what am I missing?

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1. Rounded to 1 dp, 7.3 means 7.25 ≤ L < 7.35 and 4.8 means 4.75 ≤ W < 4.85; for positive sides the area grows with both, so A ∈ [7.25×4.75, 7.35×4.85) = [34.4375, 35.6475), with the upper bound exclusive. Also, 2.40 (2 dp) means [2.395, 2.405) while 2.4 (1 dp) means [2.35, 2.45), so they’re not the same-2.40 is tighter (unless you use some quirky rounding rule).

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2. I like to picture each rounded measurement as a little “wiggle room” band, then ask how the area behaves as I slide around inside that band. For 7.3 cm to 1 dp, the true length is in [7.25, 7.35), and for 4.8 cm to 1 dp, the true width is in [4.75, 4.85). Because length and width are positive and area = length × width increases as either one increases, the smallest area happens at the bottom-left corner (7.25 × 4.75 = 34.4375 cm²), and the biggest is approached at the top-right corner (7.35 × 4.85 = 35.6475 cm²) but not actually reached because those upper bounds are exclusive. So the area interval is [34.4375, 35.6475). No need to “mix-and-match” here: with positives, lower×lower gives the minimum and upper×upper gives the supremum; the cross products sit in the middle. The only time you’d check all four corner products is when signs could flip or zero is involved. On the mini-mystery: 2.40 to 2 dp means [2.395, 2.405), while 2.4 to 1 dp means [2.35, 2.45), so they are not the same-writing that extra zero is like saying, “I measured a bit more finely,” and it really does tighten the error interval.

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3. You’re on the right track! Rounding to 1 decimal place means a half-step of 0.05, so under the usual “nearest” rounding, the true values lie in [stated − 0.05, stated + 0.05). For your rectangle: length L is 7.3 cm ⇒ 7.25 ≤ L < 7.35, and width W is 4.8 cm ⇒ 4.75 ≤ W < 4.85. Because area A = L×W is increasing in each variable for positive numbers, the extreme areas live at the “corners” of that rectangle of possibilities: the minimum is at the lower–lower corner and the supremum (the “just out of reach” maximum) at the upper–upper corner. So A_min = 7.25×4.75 = 34.4375 cm², and A is strictly less than 7.35×4.85 = 35.6475 cm². In interval language: 34.4375 ≤ A < 35.6475. The “mix-and-match” pairs land in between: for instance 7.25×4.85 = 35.1625 and 7.35×4.75 = 34.9125, both nestled between the extremes. And yes, because both upper bounds are exclusive (L < 7.35 and W < 4.85), the area’s upper bound is also exclusive: A < 7.35×4.85. You can get arbitrarily close (take L = 7.3499, W = 4.8499), but you can’t actually hit it. As for 2.40 (to 2 d.p.) versus 2.4 (to 1 d.p.), they are not the same error interval in a measurement context. Writing 2.40 to 2 decimal places means 2.395 ≤ x < 2.405, while 2.4 to 1 decimal place means 2.35 ≤ x < 2.45. So 2.40 is a tighter, more informative statement about the precision. If you’re just talking about exact decimals (no rounding implied), then 2.40 equals 2.4 exactly-same number, different outfit-but in measurement land, that extra zero matters. A nice walkthrough of error intervals and bounds lives here: https://www.corbettmaths.com/2016/12/11/error-intervals/

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