If R is the radius of a certain circle, is the NUMERICAL VALUE of the area of this circle greater than the NUMERICAL VALUE of the circumference of this circle?
1. 0 < R < 3
2. The diameter of the circle is greater than 4
\[ \]
Obs.: we cannot compare area and length, but we can compare their corresponding numerical values. That´s why I modified the question stem wording.
\[\pi {r^2}\,\,\mathop > \limits^? \,\,2\pi r\,\,\,\,\,\mathop \Leftrightarrow \limits^{:\,\,\pi r\,\, > \,\,0} \,\,\,\,\boxed{r\,\,\mathop > \limits^? \,\,2}\]
\[\left( 1 \right)\,\,\,\,\,\left\{ \begin{gathered}
\,r = 1\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,r = 3\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,2r\,\, > 4\,\,\,\, \Rightarrow \,\,\,\,r > 2\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\]
The above follows the notations and rationale taught in the GMATH method.
