The area of a circle is to be computed from a measured value of its diameter. Estimate the maximum permissible percentage error in the diameter measurement if the percentage error in the area must be kept within \( \pm 5 \% \). Enter the exact answer. Maximum Error \( = \pm \)

To find the maximum permissible percentage error in the diameter measurement, we start with the formula for the area of a circle, \( A = \pi r^2 \). The radius \( r \) is half of the diameter \( d \), so \( A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4} \). The percentage error in area (\( \Delta A/A \)) can be linked to the diameter (\( \Delta d/d \)). Using differentiation and the chain rule, we can establish: \[ \frac{\Delta A}{A} = 2 \frac{\Delta d}{d} \] We want this to be within \( \pm 5\% \) for area: \[ 2 \frac{\Delta d}{d} = 0.05 \] So: \[ \frac{\Delta d}{d} = \frac{0.05}{2} = 0.025 \] Thus, the maximum permissible percentage error in the diameter measurement is: Maximum Error \( = \pm 2.5\% \)