What is the maximum percentage error in density of body if percentage error in measurement of mass is `4%` , and percentage error in measurement of side `1%` ?

Options

A. `18%`
B. `7%`
C. `20%`
D. `21%`

Answer

To find the maximum percentage error in the density of a body given the percentage errors in mass and side measurements, we can follow these steps:

### Step 1: Understand the relationship between density, mass, and volume
Density (D) is defined as mass (M) divided by volume (V). For a cube with side length (L), the volume can be expressed as:
\[ V = L^3 \]
Thus, the density can be expressed as:
\[ D = \frac{M}{L^3} \]

### Step 2: Determine the formula for percentage error in density
To find the percentage error in density, we can use the formula for propagation of errors. The percentage error in density can be expressed in terms of the percentage errors in mass and length as follows:
\[ \frac{\Delta D}{D} \times 100 = \frac{\Delta M}{M} \times 100 + 3 \times \frac{\Delta L}{L} \times 100 \]
Where:
- \(\Delta D\) is the error in density,
- \(D\) is the density,
- \(\Delta M\) is the error in mass,
- \(M\) is the mass,
- \(\Delta L\) is the error in length,
- \(L\) is the length.

### Step 3: Substitute the given percentage errors
From the problem, we know:
- The percentage error in mass, \(\frac{\Delta M}{M} \times 100 = 4\%\)
- The percentage error in side length, \(\frac{\Delta L}{L} \times 100 = 1\%\)

### Step 4: Plug the values into the formula
Now we can substitute these values into the error formula for density:
\[ \frac{\Delta D}{D} \times 100 = 4 + 3 \times 1 \]
\[ \frac{\Delta D}{D} \times 100 = 4 + 3 = 7\% \]

### Step 5: Conclusion
The maximum percentage error in density is:
\[ \text{Maximum Percentage Error in Density} = 7\% \]