Measurement bounds in calculations

A square panel side is 10 cm to the nearest 1 cm, with halfway values rounded up. Its nominal area is 100 cm². Is every possible area within 0.5 cm² of 100 cm²?

Keep the recorded side at 10 cm and reduce its rounding unit from 1 cm to 0.2 cm. What happens to the possible area interval?

Change the side-length rounding unit. The side is recorded as 10 cm, with halfway values rounded up. The area axis rescales, so compare the numerical endpoints and width rather than the drawn width. Only rounding uncertainty is modelled

Why is the upper area boundary excluded?

Use the endpoints that make the result smallest or largest

First write each measurement interval. For positive dimensions, products increase with either input. A positive quotient increases with its numerator but decreases with its denominator

1. 1Exact scale factor Multiply both endpoints by the positive factor. Convert units afterwards
2. 2Positive product Use both lower endpoints for the lower product and both upper endpoints for the upper product
3. 3Positive quotient Use a lower numerator and upper denominator for the lower ratio. Reverse those choices for the upper ratio
4. 4If decimal bounds are required Round the lower bound down and the upper bound up. These conservative bounds may be wider than the tight interval

Plan length interval (mm)

\(\displaystyle 31.5\le l<32.5\)

Exact scale

\(\displaystyle 1:50\)

A plan length is 32 mm to the nearest 1 mm. The scale 1:50 is exact. What is the lower bound for the actual length in metres?

Multiply 31.5 mm by 50, then divide by 1000

Length interval (cm)

\(\displaystyle 19.5\le l<20.5\)

Width interval (cm)

\(\displaystyle 11.5\le w<12.5\)

A rectangular panel has length 20 cm and width 12 cm, each to the nearest 1 cm. Find its strict upper area bound

Use the upper length and width boundaries

A test vehicle travels d metres in t seconds, with 99.5 ≤ d < 100.5 and 19.5 ≤ t < 20.5. Which ratio gives the tight lower speed boundary?

Tight lower boundary

\(\displaystyle L=\frac{99.5}{20.5}\text{ m/s}\)

The tight lower speed boundary is 99.5 ÷ 20.5 m/s, approximately 4.853658… m/s. Give a conservative lower bound to 2 decimal places

Round down so the bound does not exclude possible speeds. Write exactly 2 decimal places, with no extra trailing digit

For that vehicle, the upper speed boundary is approximately 5.153846… m/s. A student rounds it to 5.15 m/s and claims v < 5.15. Is that guaranteed?

The panel area satisfies 224.25 cm² ≤ A < 256.25 cm². The design needs at least 240 cm². Does the measurement guarantee enough area?

Side interval (cm)

\(\displaystyle 15.5\le s<16.5\)

Independent check: a square side is 16 cm to the nearest 1 cm. What is the included lower bound for its area?

Square the included lower side endpoint

Evaluate the decision in writing

A rectangular panel is recorded as 20 cm by 12 cm, to the nearest 1 cm with halfway values rounded up. It must provide at least 240 cm². Write a recommendation that uses the interval, not just the nominal product

1. 1State the model Define the quantities, rounding convention and ideal-shape assumptions. Show the input and output intervals with units
2. 2Recommend a next step Explain why the present reading cannot settle the requirement. Compare finer measurement with choosing a larger panel; do not invent costs or claim a regulation requires a particular margin
3. 3Check other sources of error Explain how a ruler with a known bias or an irregular edge would affect the model. State what should be checked before using the recommendation. This written response needs teacher or reviewer feedback