Physical quantity `x` and `y` are related as `y=4tanx` .If at `x=(pi)/(4)` radian error in measurement of `x` is `2%` then find `%` error in measurement of `y` at `x=(pi)/(4)`

Options

A. `2%`
B. `(pi)/(2)%`
C. `(pi)/(6)%`
D. `pi%`

Answer

To solve the problem step by step, we need to find the percentage error in the measurement of \( y \) given the relationship \( y = 4 \tan x \) and the percentage error in \( x \).

### Step 1: Understand the relationship
We are given the relationship between \( y \) and \( x \):
\[ y = 4 \tan x \]

### Step 2: Differentiate \( y \) with respect to \( x \)
To find how changes in \( x \) affect \( y \), we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 4 \sec^2 x \]

### Step 3: Express the differential change in \( y \)
Using the differentiation result, we can express the differential change in \( y \):
\[ dy = 4 \sec^2 x \, dx \]

### Step 4: Find the value of \( x \) and \( y \) at \( x = \frac{\pi}{4} \)
At \( x = \frac{\pi}{4} \):
\[ y = 4 \tan\left(\frac{\pi}{4}\right) = 4 \cdot 1 = 4 \]
Also, we find \( \sec^2\left(\frac{\pi}{4}\right) \):
\[ \sec^2\left(\frac{\pi}{4}\right) = 2 \]

### Step 5: Substitute back into the differential equation
Now substituting \( \sec^2\left(\frac{\pi}{4}\right) \) into the equation for \( dy \):
\[ dy = 4 \cdot 2 \, dx = 8 \, dx \]

### Step 6: Calculate the percentage error in \( y \)
The percentage error in \( y \) can be calculated using the formula:
\[ \frac{dy}{y} = \frac{8 \, dx}{4} = 2 \frac{dx}{4} = 2 \cdot \frac{dx}{y} \]
Given that the percentage error in \( x \) is \( 2\% \), we have:
\[ \frac{dx}{x} = 0.02 \]

### Step 7: Substitute the percentage error in \( x \)
Thus, substituting this into the equation gives:
\[ \frac{dy}{y} = 2 \cdot 0.02 = 0.04 \]

### Step 8: Convert to percentage
To convert this to a percentage:
\[ \text{Percentage error in } y = 0.04 \times 100\% = 4\% \]

### Conclusion
The percentage error in the measurement of \( y \) at \( x = \frac{\pi}{4} \) is:
\[ \boxed{4\%} \]