It seems you're asking about the uncertainty of an expression that involves \( A = 1 / (4 \pi d^2) \), where \( d \) is likely a distance. Let's clarify how to calculate the uncertainty of \( A \) given the uncertainty in \( d \).

### The equation:
\[
A = \frac{1}{4 \pi d^2}
\]

### Uncertainty propagation:
To find the uncertainty of \( A \), \( \Delta A \), we use the propagation of uncertainty formula. If the variable \( A \) depends on another variable \( d \), then the uncertainty in \( A \) is related to the uncertainty in \( d \) by:

\[
\frac{\Delta A}{A} = \left| \frac{dA}{d d} \right| \frac{\Delta d}{d}
\]

First, let's compute the derivative \( \frac{dA}{d d} \):

\[
A = \frac{1}{4 \pi d^2}
\]
Taking the derivative of \( A \) with respect to \( d \):
\[
\frac{dA}{d d} = - \frac{2}{4 \pi d^3} = - \frac{1}{2 \pi d^3}
\]

Now, applying the uncertainty propagation formula:
\[
\frac{\Delta A}{A} = \left| - \frac{1}{2 \pi d^3} \right| \cdot \frac{\Delta d}{d} = 2 \frac{\Delta d}{d}
\]

### Final result:
\[
\Delta A = 2 \frac{\Delta d}{d} A
\]
So the uncertainty in \( A \), \( \Delta A \), is proportional to \( 2 \frac{\Delta d}{d} \).

### Example:
If the relative uncertainty in \( d \) is 1%, the relative uncertainty in \( A \) would be 2%. You just need to plug in the values of \( d \) and \( \Delta d \) to get the final uncertainty.

Would you like to go through a specific example with values?

### 5 Related Questions:
1. How do you calculate the uncertainty in a function of multiple variables?
2. What is the relative uncertainty, and how is it useful?
3. How can we minimize uncertainties in measurements?
4. What is the difference between absolute and relative uncertainty?
5. How do uncertainty and error differ in experimental physics?

### Tip:
Always double-check units when calculating uncertainties to ensure consistency throughout your calculations!

Learn how to calculate the uncertainty in the equation A = 1 / (4πd^2) using uncertainty propagation. This example covers differentiation and the use of the uncertainty propagation formula, ideal for college-level physics or advanced calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation