Question

The density of the material of a cube can be estimated by measuring its mass and the length of one of its sides. If the maximum error in the measurement of mass and length are 0.3% and 0.2% respectively, the maximum error in the estimation of the density of the cube is approximately

• A. 0.011
• B. 0.005
• C. 0.009
• D. 0.007

Answer 1

The Correct Option is C

To determine the maximum error in the estimation of the density of the cube, we start by understanding the concept of errors in measurements and how they propagate through calculations.

The density \(\rho\) of a cube is given by the formula:

\rho = \frac{m}{V}

where \(m\) is the mass and \(V\) is the volume of the cube. The volume \(V\) of a cube with side length \(l\) is:

V = l^3

Thus, the density can be rewritten as:

\rho = \frac{m}{l^3}

To find the maximum error in density, we need to use the error propagation formulas:

The relative error in a product or quotient is given by the sum of the relative errors in the individual measurements. Therefore, the relative error in density is:

\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \times \frac{\Delta l}{l}

where:

• \frac{\Delta m}{m} is the relative error in mass, which is given as 0.3% or 0.003.
• \frac{\Delta l}{l} is the relative error in length, which is given as 0.2% or 0.002.

Substituting the values, we have:

\frac{\Delta \rho}{\rho} = 0.003 + 3 \times 0.002

\frac{\Delta \rho}{\rho} = 0.003 + 0.006 = 0.009

Thus, the maximum error in the estimation of the density of the cube is approximately 0.009 or 0.9%.

Therefore, the correct answer is 0.009.