The formula for the area of a circle is given by \(A = \pi r^2\), where \(r\) is the radius of the circle. If the radius is given as \(5\) cm subject to an error of \(0.1\) cm, then the maximum error in the area of the circle can be calculated using differentiation.

Differentiating both sides of the equation with respect to \(r\), we get \(\frac{dA}{dr} = 2\pi r\). Substituting \(r = 5\) cm, we get \(\frac{dA}{dr} = 10\pi\).

The maximum error in the area of the circle is given by \(\Delta A = \left|\frac{dA}{dr}\right| \Delta r = 10\pi \times 0.1 = \pi\) cm².

The percentage error in the area of the circle is given by \(\frac{\Delta A}{A} \times 100\%\). Substituting the values for \(\Delta A\) and \(A = \pi r^2 = 25\pi\) cm², we get \(\frac{\Delta A}{A} \times 100\% = \frac{\pi}{25\pi} \times 100\% = 4\%\).