The density of a sphere is measured by measuring the mass and diameter. If it is known that the maximum percentage errors in measurement of mass and diameter are `2%` and `3%` respectively then the maximum percentage error in the measurement of density is

Answer

To find the maximum percentage error in the measurement of density, we can follow these steps:

### Step 1: Understand the formula for density
The density \( \rho \) 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 is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Since the diameter \( d \) is related to the radius \( r \) by \( d = 2r \), we can express the volume in terms of diameter:
\[ V = \frac{4}{3} \pi \left(\frac{d}{2}\right)^3 = \frac{4}{3} \pi \frac{d^3}{8} = \frac{\pi d^3}{6} \]

### Step 3: Substitute the volume into the density formula
Substituting the volume into the density formula gives:
\[ \rho = \frac{m}{\frac{\pi d^3}{6}} = \frac{6m}{\pi d^3} \]

### Step 4: Determine the percentage error in density
The percentage error in density can be calculated using the formula for propagation of errors. If \( \Delta M \) is the error in mass and \( \Delta D \) is the error in diameter, the percentage error in density \( \Delta \rho \) is given by:
\[ \frac{\Delta \rho}{\rho} \times 100 = \frac{\Delta M}{M} \times 100 + 3 \times \frac{\Delta D}{D} \times 100 \]
Here, the factor of 3 comes from the \( d^3 \) term in the denominator, since the error in diameter affects the volume cubically.

### Step 5: Substitute the given percentage errors
Given that the maximum percentage error in mass \( \frac{\Delta M}{M} \times 100 \) is \( 2\% \) and the maximum percentage error in diameter \( \frac{\Delta D}{D} \times 100 \) is \( 3\% \), we can substitute these values into the equation:
\[ \text{Percentage error in density} = 2\% + 3 \times 3\% \]

### Step 6: Calculate the total percentage error
Calculating this gives:
\[ \text{Percentage error in density} = 2\% + 9\% = 11\% \]

### Final Answer
Thus, the maximum percentage error in the measurement of density is:
\[ \boxed{11\%} \]