If the error in measuring the radius of the sphere is 2% and that in measuring its mass is 3%, Then the error in measuring the density of material of the sphere is:

Options

A. 0.05
B. 0.07
C. 0.09
D. 0.11

Explanation

To find the error in measuring the density of a sphere given the errors in measuring its radius and mass, we can follow these steps:

### Step 1: Understand the relationship between mass, volume, and density
The density (ρ) of a sphere is defined as:
\[ \rho = \frac{m}{V} \]
where \( m \) is the mass and \( V \) is the volume of the sphere.

### Step 2: Write the formula for the volume of a sphere
The volume \( V \) of a sphere with radius \( r \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the relative error in volume
To find the relative error in volume, we need to differentiate the volume formula:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
where \( \Delta r \) is the error in measuring the radius.

### Step 4: Substitute the given errors
We are given:
- The error in measuring the radius \( \frac{\Delta r}{r} = 2\% = 0.02 \)
- The error in measuring the mass \( \frac{\Delta m}{m} = 3\% = 0.03 \)

Now, substituting the error in radius into the volume error formula:
\[ \frac{\Delta V}{V} = 3 \times 0.02 = 0.06 \text{ or } 6\% \]

### Step 5: Calculate the total relative error in density
The relative error in density can be calculated using the formula:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta V}{V} \]
Substituting the errors we have:
\[ \frac{\Delta \rho}{\rho} = 0.03 + 0.06 = 0.09 \text{ or } 9\% \]

### Final Answer
The error in measuring the density of the material of the sphere is **9%**.

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