A metal box is heated until each of its sides has expanded by 0.1%. By what percent has the volume of the box changed?

Question

BlackTom AI · Accepted Answer

We start by recognizing that the problem deals with thermal expansion in three dimensions. If each linear dimension increases by 0.1%, the fractional increase in each dimension is 0.001 (since 0.1% = 0.001 in decimal).

Option 1: '+
0.1
%' — This would imply the volume increases by 0.1%. However, the volume depends on all three dimensions, so the total fractional change in volume is approximately three times the fractional change in a single dimension for small expansions, not just the change in one dimension. Therefore, this underestimates the true volume change.

Option 2: '+
0.3
%' — This matches the first-order approximation for the volume change when each linear dimension expands by the same small fraction. Since volume scales as (1+0.001)^3 ≈ 1 + 3×0.001 = 1.003, the fractional change is about 0.3%. This option aligns with the expected three-dimensional expansion behavior.

Option 3: '−
0.3
%' — This would indicate a decrease in volume, which contradicts the fact that heating increases all linear dimensions. Therefore, a negative volume change is incorrect.

Option 4: '−
0.1
%' — This expresses a decrease of 0.1% in volume, which again contradicts the expansion of all sides. Since all sides lengthen, the volume cannot shrink in this scenario.

In sum, the physically correct interpretation for small, uniform linear expansions in three dimensions yields a positive volume change around 0.3%, while the other options either misstate the amount or imply a contraction that does not occur under heating.

SPHY001 Practice Quiz - Unit 13 | LA053

A metal box is heated until each of its sides has expanded by 0.1%. By what percent has the volume of the box changed?

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