A physical quantity 'y' is represented by the formula `y=m^(2)r^(-4)g^(x)l^(-(3)/(2))`
If the percentage errors found in y, m, r, l and g are 18, 1, 0.5, 4 and p respectively, then find the value of x and p.

Options

A. 5 and `pm2`
B. 4 and `pm3`
C. `(16)/(3)` and `pm (3)/(2)`
D. 8 and `pm2`

Answer

To solve the problem, we need to analyze the given formula for the physical quantity \( y \) and the associated percentage errors. The formula is:

\[ y = m^2 r^{-4} g^x l^{-\frac{3}{2}} \]

### Step 1: Write the formula for percentage error in \( y \)

The percentage error in \( y \) can be expressed in terms of the percentage errors in \( m \), \( r \), \( g \), and \( l \) using the following relationship:

\[ \text{Percentage Error in } y = \left| \frac{\Delta y}{y} \right| \times 100 \]

### Step 2: Express the relative errors

Using the formula for \( y \), we can express the relative error in \( y \) as:

\[ \frac{\Delta y}{y} = 2 \frac{\Delta m}{m} + 4 \frac{\Delta r}{r} + x \frac{\Delta g}{g} + \frac{3}{2} \frac{\Delta l}{l} \]

### Step 3: Substitute the given percentage errors

The given percentage errors are:
- \( \Delta m/m = 1\% \)
- \( \Delta r/r = 0.5\% \)
- \( \Delta g/g = p\% \)
- \( \Delta l/l = 4\% \)

Substituting these values into the equation:

\[ \frac{\Delta y}{y} = 2 \times 1 + 4 \times 0.5 + x \times p + \frac{3}{2} \times 4 \]

### Step 4: Simplify the equation

Calculating each term:

- \( 2 \times 1 = 2 \)
- \( 4 \times 0.5 = 2 \)
- \( \frac{3}{2} \times 4 = 6 \)

Thus, the equation becomes:

\[ \frac{\Delta y}{y} = 2 + 2 + x \times p + 6 \]

This simplifies to:

\[ \frac{\Delta y}{y} = 10 + x \times p \]

### Step 5: Set the equation equal to the given percentage error in \( y \)

We know the percentage error in \( y \) is given as \( 18\% \):

\[ 10 + x \times p = 18 \]

### Step 6: Solve for \( x \times p \)

Rearranging gives:

\[ x \times p = 18 - 10 = 8 \]

### Step 7: Determine possible values for \( x \) and \( p \)

Now we need to find pairs of \( (x, p) \) such that their product is \( 8 \). The possible pairs of integers could be:

- \( (1, 8) \)
- \( (2, 4) \)
- \( (4, 2) \)
- \( (8, 1) \)

### Step 8: Conclusion

Since we need to find values of \( x \) and \( p \) that satisfy \( x \times p = 8 \), we can choose:

- \( x = 2 \) and \( p = 4 \) (which satisfies the condition).

Thus, the final values are:

\[ x = 2, \quad p = 4 \]