What is the percentage error in the physical quantity A If it is related to four other physical quantities a, b, c and d as `A=(a^(2)b^(3))/(d sqrt(c))`? The percentage error in measurement of a, b, c and d are1%, 3%, 2% and 1% respectively.

Explanation

To find the percentage error in the physical quantity \( A \) given by the relation

\[ A = \frac{a^2 b^3}{d \sqrt{c}}, \]

we will use the rules for calculating percentage errors in products and powers of quantities.

### Step 1: Identify the powers of each variable in the equation for \( A \)

From the equation, we can see that:
- \( a \) has a power of 2,
- \( b \) has a power of 3,
- \( c \) has a power of \( -\frac{1}{2} \) (since it's in the denominator as \( \sqrt{c} \)),
- \( d \) has a power of -1 (since it's also in the denominator).

### Step 2: Write the formula for percentage error

The general formula for the percentage error in a quantity \( A \) that depends on several other quantities is given by:

\[ \text{Percentage Error in } A = \left| m \cdot \frac{\Delta a}{a} \right| + \left| n \cdot \frac{\Delta b}{b} \right| + \left| p \cdot \frac{\Delta c}{c} \right| + \left| q \cdot \frac{\Delta d}{d} \right|, \]

where \( m, n, p, q \) are the powers of \( a, b, c, d \) respectively.

### Step 3: Substitute the powers and the given percentage errors

Given the percentage errors:
- \( \frac{\Delta a}{a} = 1\% = 0.01 \)
- \( \frac{\Delta b}{b} = 3\% = 0.03 \)
- \( \frac{\Delta c}{c} = 2\% = 0.02 \)
- \( \frac{\Delta d}{d} = 1\% = 0.01 \)

Now substituting the values into the formula:

\[ \text{Percentage Error in } A = 2 \cdot 0.01 + 3 \cdot 0.03 + \left(-\frac{1}{2}\right) \cdot 0.02 + (-1) \cdot 0.01. \]

### Step 4: Calculate each term

Calculating each term:
- For \( a \): \( 2 \cdot 0.01 = 0.02 \)
- For \( b \): \( 3 \cdot 0.03 = 0.09 \)
- For \( c \): \( -\frac{1}{2} \cdot 0.02 = -0.01 \)
- For \( d \): \( -1 \cdot 0.01 = -0.01 \)

### Step 5: Sum the contributions

Now summing these contributions:

\[ \text{Percentage Error in } A = 0.02 + 0.09 - 0.01 - 0.01 = 0.09. \]

### Step 6: Convert to percentage

To convert to percentage, we multiply by 100:

\[ \text{Percentage Error in } A = 0.09 \times 100 = 9\%. \]

Thus, the percentage error in the physical quantity \( A \) is **9%**.

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