Error Propagation

Question

While determining density, a metal cube has its mass measured with 2% error and the length of each side measured with 1% error in an experiment. What is the maximum percentage error in the calculated density?

RankGuru · Accepted Answer

According to NCERT Class 11, Chapter 2 (Units and Measurements), when a quantity is computed from a product or quotient of measured quantities, the maximum relative errors add, with each error multiplied by the power of that quantity in the formula. Density is mass divided by volume, and volume of a cube is the side cubed, so density = m / L^3. The fractional error therefore equals the error in mass plus three times the error in length: 2% + 3 × 1% = 2% + 3% = 5%. The option 3% is wrong because it forgets to multiply the length error by the power three. The option 7% is wrong as it incorrectly adds 2% to 5% or mis-counts the powers. The option 4% is wrong because it uses a power of two instead of three for the volume term. A sanity check confirms that since volume depends strongly on length cubed, the length error must dominate the propagation, which our result reflects.

Error Propagation

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