Question

Error in the measurement of radius of the sphere is 2%. The error in the calculated value of its volume is

• A. 3%
• B. 2%
• C. 6%
• D. 9%

Hint:

When calculating the error in a derived quantity, use the power rule for relative errors. For example, if a quantity is proportional to \( r^n \), then the relative error is \( n \times \) (relative error in \( r \)).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the percentage error in volume based on the percentage error in the radius measurement.
Step 2: Key Formula or Approach:
The volume of a sphere is \( V = \frac{4}{3} \pi r^3 \).
For small errors, the relative error in \( V \) is given by:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
Step 3: Detailed Explanation:
Given percentage error in radius: \( \frac{\Delta r}{r} \times 100% = 2% \).
Applying the formula for relative error:
\[ % \text{ error in } V = \left( \frac{\Delta V}{V} \times 100% \right) = 3 \times \left( \frac{\Delta r}{r} \times 100% \right) \]
\[ % \text{ error in } V = 3 \times 2% = 6% \]
Step 4: Final Answer:
The error in the calculated volume is \( 6% \).