The pressure on a circular plate is measured by measuring the force on the plate and the radius of the plate. If the errors in measurement of force and radius are 5% and 3% , respectively , then the percentage of error in the measurement of pressure is

Options

A. 8
B. 14
C. 11
D. 12

Answer

To solve the problem of finding the percentage error in the measurement of pressure based on the errors in the measurements of force and radius, we can follow these steps:

### Step 1: Understand the formula for pressure
Pressure (P) is defined as the force (F) applied per unit area (A). For a circular plate, the area can be calculated using the formula:
\[ A = \pi r^2 \]
Thus, the formula for pressure becomes:
\[ P = \frac{F}{A} = \frac{F}{\pi r^2} \]

### Step 2: Identify the variables and their errors
- The error in the measurement of force (F) is given as 5%.
- The error in the measurement of radius (r) is given as 3%.

### Step 3: Determine the relationship between the errors
Using the formula for pressure, we can see that:
- The power of F in the formula is 1 (since it is in the numerator).
- The power of r in the formula is -2 (since it is squared in the denominator).

### Step 4: Calculate the percentage error in pressure
The formula for the percentage error in a function of multiple variables can be expressed as:
\[ \text{Percentage Error in } P = \left( \frac{dF}{F} \times 100 \right) + \left( \left( -2 \frac{dr}{r} \right) \times 100 \right) \]

Substituting the given errors:
- For force (F): \( \frac{dF}{F} \times 100 = 5\% \)
- For radius (r): \( \frac{dr}{r} \times 100 = 3\% \)

Thus, the percentage error in pressure becomes:
\[ \text{Percentage Error in } P = 5 + 2 \times 3 \]
\[ = 5 + 6 \]
\[ = 11\% \]

### Final Answer
The percentage error in the measurement of pressure is **11%**.

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