Question

If the percentage error in the radius of a circle is 3, then the percentage error in its area is:

• A. \( 6 \)
• B. \( \frac{3}{2} \)
• C. \( 2 \)
• D. \( 4 \)

Hint:

For percentage errors in power functions, multiply the error in the variable by the exponent.

Answer 1

The Correct Option is A

Step 1: Understanding percentage error propagation The area of a circle is given by: \[ A = \pi r^2. \] Differentiating both sides: \[ \frac{dA}{A} = 2 \frac{dr}{r}. \] Multiplying by 100 to convert to percentage error: \[ \% \text{ error in } A = 2 \times (\% \text{ error in } r). \] Step 2: Substituting given values Given \( \% \) error in \( r = 3 \): \[ \% \text{ error in } A = 2 \times 3 = 6. \]

Answer 2

Given:
Percentage error in radius \( r \) of a circle is 3%.

Step 1: Recall the formula for the area of a circle:
\[ A = \pi r^2 \]

Step 2: The percentage error in area when radius has a percentage error \( \delta r \) is:
\[ \delta A = 2 \times \delta r \]

Step 3: Substitute \( \delta r = 3\% \):
\[ \delta A = 2 \times 3\% = 6\% \]

Therefore, the percentage error in the area is:
\[ \boxed{6\%} \]