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The percentage error in measuring the side of a cube is 2%. Then the error in its volume is-


1. 2%
2. 3%
3. 6%
4. 1.5%

1 Answer

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Best answer
Correct Answer - Option 3 : 6%

The correct answer is option 3) i.e. 6%

Concept:

The error of a product or a quotient: If ​two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors, the possible error ΔZ in the operation Z = AB or Z = A ÷ B is given by:
\( \frac{\Delta Z}{ Z} = \frac{∆A}{A} + \frac{∆B}{B}\)

Calculation:

Let the side of a cube be 's'.

The volume of the cube, V = s3

Given that, 

\(\frac{\Delta s}{s} = 2\% =0.02\)

On differentiating V w.r.t s, 

\(\frac{\Delta V}{\Delta s} = 3s^2\)

\(\Rightarrow \Delta V= {3s^2}\times \Delta s\)

\(\Rightarrow \frac{\Delta V}{V} = \frac{3s^2}{s^3} \times\Delta s\)

\(\Rightarrow \frac{\Delta V}{V} = 3\times\frac{\Delta s}{s}\)

\(\Rightarrow \frac{\Delta V}{V} = 3\times 0.02 = 0.06 = 6\%\)

Combination of errors: 

When physical quantities involving a mathematical operation have errors associated with them, there will be an error in the result arising from its combination.

1. The error of a sum or a difference: If ​two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors, the possible error ΔZ in the operation Z = A ± B is given by:
± ΔZ = ± ΔA ± ΔB

2. The error in case of a measured quantity raised to a power: ​
If Z = Ax then 

\(\frac{\Delta Z}{Z} = x\frac{\Delta A}{A}\)

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