A physical quantity `rho` is calculated by using the formula `rho =(1)/(10)(xy^(2))/(z^(1//3))`, where x, y and z are experimentally measured quantities. If the fractional error in the measurement of x, y and z are `2%, 1%` and `3%`, respectively, then the maximum fractional error in the calculation of `rho` is

Options

A. `0.5 % `
B. `5%`
C. `6%`
D. `7%`

Explanation

To find the maximum fractional error in the calculation of the physical quantity \( \rho \), we start with the given formula:

\[ \rho = \frac{1}{10} \frac{xy^2}{z^{1/3}} \]

### Step 1: Identify the formula for fractional error
The fractional error in a physical quantity calculated from measured quantities can be determined using the following rules:
- For multiplication and division, the fractional errors add up.
- For powers, the fractional error is multiplied by the exponent.

### Step 2: Write the expression for fractional error in \( \rho \)
From the formula, we can express the fractional error in \( \rho \) as:

\[ \frac{\Delta \rho}{\rho} = \frac{\Delta x}{x} + 2\frac{\Delta y}{y} + \frac{1}{3}\frac{\Delta z}{z} \]

### Step 3: Substitute the given fractional errors
We know the fractional errors for \( x \), \( y \), and \( z \):
- Fractional error in \( x \) = 2% = 0.02
- Fractional error in \( y \) = 1% = 0.01
- Fractional error in \( z \) = 3% = 0.03

Substituting these values into the expression:

\[ \frac{\Delta \rho}{\rho} = 0.02 + 2(0.01) + \frac{1}{3}(0.03) \]

### Step 4: Calculate the contributions
Calculating each term:

1. \( 0.02 \) (from \( x \))
2. \( 2(0.01) = 0.02 \) (from \( y^2 \))
3. \( \frac{1}{3}(0.03) = 0.01 \) (from \( z^{1/3} \))

Now, add these contributions together:

\[ \frac{\Delta \rho}{\rho} = 0.02 + 0.02 + 0.01 = 0.05 \]

### Step 5: Convert to percentage
To express this as a percentage, we multiply by 100:

\[ \text{Percentage error in } \rho = 0.05 \times 100 = 5\% \]

### Final Answer
The maximum fractional error in the calculation of \( \rho \) is **5%**.

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