91Ó°ÊÓ

The given radius of the circle is $3.1×{10}^4\mathrm{cm}$.

The area of a circle represents the region covered by a circular object in the 2-D plane. In the standard international system, the area of a circle is measured in meter square.

The relation for the area of a circle is $A=\mathrm{Ï€}{r}^2$.

Substituting the variables by their values in the above relation, you get

$\begin{array}{c}A=\mathrm{π}{\left(3.1×{10}^4\mathrm{cm}\right)}^2\\ =3.01×{10}^9{\mathrm{cm}}^2.\end{array}$

Uncertainty represents the range within which the actual value of measurement lies.

The relation for uncertainty in the area is

$\frac{\mathrm{Δ}A}{A}=\frac{2\left(\mathrm{Δ}r\right)}{r}.$

Here, $\mathrm{Δ}r$ is the smallest possible uncertainty of the radius with the value $0.1×{10}^4\mathrm{cm}$.

Substitute the variables by their values in the above relation, you get

$\begin{array}{c}\frac{\mathrm{Δ}A}{A}=\left[\frac{2\left(0.1×{10}^4\mathrm{cm}\right)}{\left(3.1×{10}^4\mathrm{cm}\right)}\right]\\ ⇒\frac{\mathrm{Δ}A}{A}=0.0645×100\%\\ ⇒\frac{\mathrm{Δ}A}{A}=6.45\%.\end{array}$

Thus, the area of the circle and the approximation uncertainly are $A=3.01×{10}^9{\mathrm{cm}}^2$and $\frac{\mathrm{Δ}A}{A}=6.45\%$, respectively.