Question

The error in the measurement of length and mass is 3% and 4% respectively. The error in the measurement of density will be ______.

• A. 6%
• B. 13%
• C. 9%
• D. 15%

Hint:

In error analysis, NEVER subtract errors, even if variables are divided (like $M/L^3$)! We always want the MAXIMUM possible error, so we must add them up.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the percentage errors for mass and length measurements of an object (implied to be a cube or a sphere whose volume depends on length cubed). We must find the maximum percentage error in the calculated density.

Step 2: Detailed Explanation:
The formula relating density ($\rho$), mass ($M$), and volume ($V$) is:
$\rho = \frac{M}{V}$
For a regular 3D solid (like a cube of side $L$ or a sphere of radius $L$), volume is proportional to the cube of its linear dimension:
$V \propto L^3$
Therefore, the density equation becomes:
$\rho = \text{constant} \times \frac{M}{L^3}$
When calculating the maximum relative percentage error, powers are brought down as simple multipliers, and all individual errors are strictly ADDED together (because errors always compound):
$\frac{\Delta \rho}{\rho} \times 100 = \left( \frac{\Delta M}{M} \times 100 \right) + 3 \left( \frac{\Delta L}{L} \times 100 \right)$
We are given the individual percentage errors:
Percentage error in mass $\left( \frac{\Delta M}{M} \times 100 \right) = 4%$
Percentage error in length $\left( \frac{\Delta L}{L} \times 100 \right) = 3%$
Substitute these values into the error formula:
$%\text{ Error in density} = 4% + 3 \times (3%)$
$%\text{ Error in density} = 4% + 9%$
$%\text{ Error in density} = 13%$

Step 3: Final Answer:
The error is 13%, matching option (b).