A volume claim based on rounded data

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Determinants and singular matrices. Read the relevant method, then decide which parts you can solve without hints.

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Read an original example question

A three-dimensional transformation has diagonal matrix diag(1, 1, a), where a is 3 to the nearest integer. A solid has volume 13.0 cm³ to the nearest 0.1 cm³.

Part a

State the intervals containing a and the original volume.

Part b

Find the least upper bound for the transformed volume.

Part c

A student uses 39 cm³ as an exact transformed volume. Comment.

Model solution and review criteria

Part a

2.5 ≤ a < 3.5; 12.95 ≤ V < 13.05.

Use half a unit and half a tenth.
Keep the upper boundaries open.

Part b

Positive factors preserve order. The transformed volume is aV, with least upper bound (3.5)(13.05) = $\frac{1827}{40}$ .

Use determinant a as the volume scale.
Multiply upper bounds, not rounded central values.

Part c

The product 39 uses rounded central values. The true volume can be larger or smaller; only an interval is justified. The least upper bound is not attained by these half-open inputs.

Explain the effect of rounding.
Distinguish an upper bound from an attained maximum.

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