In an experiment, a physical quantity is given by `Y=(a^(2)b)/(c^(3))`. The permissible percentage error

Options

A. `((Delta a)/(a)+(Delta b)/(b)+(Delta c)/(c ))xx100`
B. `(2.(Delta a)/(a)+(Delta b)/(b)-(3Delta c)/(c ))xx100`
C. `(2.(Delta a)/(a)+(Delta b)/(b)+(3Delta c)/(c ))xx100`
D. `(2.(Delta a)/(a)-(Delta b)/(b)-(3Delta c)/(c ))xx100`

Answer

To solve the problem of determining the permissible percentage error in the physical quantity \( Y = \frac{a^2 b}{c^3} \), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for \( Y \)**:
We have the physical quantity defined as:
\[ Y = \frac{a^2 b}{c^3} \]

2. **Understand the concept of errors**:
In measurements, we denote the measured values as:
- \( a \) is measured as \( a + \Delta a \)
- \( b \) is measured as \( b + \Delta b \)
- \( c \) is measured as \( c + \Delta c \)
where \( \Delta a \), \( \Delta b \), and \( \Delta c \) are the absolute errors in the measurements of \( a \), \( b \), and \( c \) respectively.

3. **Calculate the fractional error in \( Y \)**:
The fractional error in \( Y \) can be derived from the formula for \( Y \):
- The fractional error due to \( a \) (since \( a \) is squared):
\[ \frac{\Delta Y_a}{Y} = 2 \cdot \frac{\Delta a}{a} \]
- The fractional error due to \( b \):
\[ \frac{\Delta Y_b}{Y} = \frac{\Delta b}{b} \]
- The fractional error due to \( c \) (since \( c \) is cubed):
\[ \frac{\Delta Y_c}{Y} = 3 \cdot \frac{\Delta c}{c} \]

4. **Combine the fractional errors**:
Since \( c \) is in the denominator, its contribution to the error will be negative:
\[ \frac{\Delta Y}{Y} = 2 \cdot \frac{\Delta a}{a} + \frac{\Delta b}{b} - 3 \cdot \frac{\Delta c}{c} \]

5. **Convert to percentage error**:
The percentage error in \( Y \) is given by:
\[ \text{Percentage Error} = \left( \frac{\Delta Y}{Y} \right) \times 100 \]
Substituting the expression for fractional error:
\[ \text{Percentage Error} = \left( 2 \cdot \frac{\Delta a}{a} + \frac{\Delta b}{b} - 3 \cdot \frac{\Delta c}{c} \right) \times 100 \]

6. **Final expression**:
The final expression for the permissible percentage error in \( Y \) is:
\[ \text{Percentage Error} = 2 \cdot \frac{\Delta a}{a} \times 100 + \frac{\Delta b}{b} \times 100 - 3 \cdot \frac{\Delta c}{c} \times 100 \]