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 problem.
The problem involves a sphere, and the error in the measurement of its radius is given as 2%. We need to find the error in the calculated volume of the sphere. The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3, \]
where \( r \) is the radius of the sphere.

Step 2: Relating the error in radius to the error in volume.
The error in the volume of the sphere can be calculated by using the concept of relative errors. We know that the volume of the sphere depends on \( r^3 \), so the error in the volume is related to the error in the radius by the following formula:
\[ \text{Relative error in volume} = 3 \times \text{Relative error in radius}. \]
This is because the volume is proportional to the cube of the radius.

Step 3: Calculating the error in volume.
We are given that the error in the radius is 2%, which means the relative error in radius is 2%. Using the relationship from Step 2, the error in the volume is:
\[ \text{Relative error in volume} = 3 \times 2% = 6%. \]

Step 4: Conclusion.
Thus, the error in the calculated volume of the sphere is 6%.
Final Answer:
The error in the volume is:
\[ \boxed{6%}. \]