Consider the ratio `r=(((1-a))/((1+a)))` to be determined by measuring a dimensionless quantity a. If the error in the measurement of a is `triangle a(triangle a"/"a lt lt 1)`, then what is the error `triangle r` in determining r?

Options

A. `(triangle a)/((1+a)^2)`
B. `(2triangle a)/((1+a)^2)`
C. `(2triangle a)/((1-a)^2)`
D. `(2a triangle a)/((1-a)^2)`

Explanation

To find the error \(\Delta r\) in determining the ratio \(r = \frac{1-a}{1+a}\) when measuring the dimensionless quantity \(a\) with an error \(\Delta a\), we can follow these steps:

### Step 1: Define the ratio
We start with the ratio given in the problem:
\[ r = \frac{1-a}{1+a} \]

### Step 2: Differentiate \(r\) with respect to \(a\)
To find the error in \(r\), we need to differentiate \(r\) with respect to \(a\):
\[ \frac{dr}{da} = \frac{(1+a)(-1) - (1-a)(1)}{(1+a)^2} \]

### Step 3: Simplify the derivative
Now, let's simplify the derivative:
\[ \frac{dr}{da} = \frac{-(1+a) - (1-a)}{(1+a)^2} = \frac{-1 - a - 1 + a}{(1+a)^2} = \frac{-2}{(1+a)^2} \]

### Step 4: Relate the error in \(r\) to the error in \(a\)
The error in \(r\) can be expressed in terms of the error in \(a\):
\[ \Delta r = \left| \frac{dr}{da} \right| \Delta a \]
Substituting the derivative we found:
\[ \Delta r = \left| \frac{-2}{(1+a)^2} \right| \Delta a = \frac{2 \Delta a}{(1+a)^2} \]

### Final Result
Thus, the error \(\Delta r\) in determining \(r\) is:
\[ \Delta r = \frac{2 \Delta a}{(1+a)^2} \]

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