Source: Extra Practice

The density of a solid cube is determined by measuring its mass and the length of its side. If the maximum error in the measurement of mass is 3%3\% and the maximum error in the measurement of length is 2%2\%, what is the maximum percentage error in the calculated density?

Options

Option A

5%

Option B

7%

Option C is correct

9%

Option D

11%

Explanation

Density ρ\rho is defined as ρ=MassVolume\rho = \frac{Mass}{Volume}. For a cube of side LL, the volume V=L3V = L^3. Thus, ρ=ML3\rho = \frac{M}{L^3}. The relative error in density is given by Δρρ=ΔMM+3ΔLL\frac{\Delta \rho}{\rho} = \frac{\Delta M}{M} + 3\frac{\Delta L}{L}. Converting this to percentage error: Percentage error in ρ=(Percentage error in M)+3×(Percentage error in L)\rho = (\text{Percentage error in } M) + 3 \times (\text{Percentage error in } L). Substituting the values: 3%+3(2%)=3%+6%=9%3\% + 3(2\%) = 3\% + 6\% = 9\%. Option 5% is incorrect because it simply adds the errors without accounting for the cubic relationship of length. Option 7% is incorrect calculation, and 11% is irrelevant.