Radius of a sphere is measured with 5% uncertainty. What is the uncertainty in the volume, determined from this radius?

Correct option is D

​​Solution:

To find the uncertainty in the volume of a sphere based on the uncertainty in its radius, we can use the concept of propagation of uncertainty.

The formula for the volume VVV of a sphere is: $$V = \frac{4}{3} \pi r^3$$

Where:

• $V$ is the volume of the sphere,
• $r$ is the radius of the sphere.

Step 1: Uncertainty Propagation

The uncertainty in the volume can be related to the uncertainty in the radius using the following formula: $$\frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r}$$

Where:

• $\Delta V$ is the uncertainty in the volume,
• $V$ is the volume of the sphere,
• $\Delta r$ is the uncertainty in the radius,
• $r$ is the radius of the sphere.

Step 2: Given Uncertainty in Radius

The problem states that the uncertainty in the radius is 5%, which means: $$\frac{\Delta r}{r} = 0.05$$

Step 3: Calculate Uncertainty in Volume

Using the formula for propagation of uncertainty: $$\frac{\Delta V}{V} = 3 \cdot 0.05 = 0.15$$

Thus, the uncertainty in the volume of the sphere is 15%.

Final Answer: The uncertainty in the volume of the sphere is 15%.

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