The numerical value of the area of a circle is greater than the numerical value of its circumference. Is this statement true? Why?
(B) The statement is false.
Let the radius of the circle be $r$.
The area of the circle is $A = \pi r^2$.
The circumference of the circle is $C = 2\pi r$.
Comparing the two,we look at the inequality $\pi r^2 > 2\pi r$.
Dividing both sides by $\pi r$ (assuming $r > 0$),we get $r > 2$.
Therefore,the area is greater than the circumference only when the radius $r > 2$.
If $0 < r < 2$,the circumference is greater than the area.
If $r = 2$,the numerical values are equal.
Thus,the statement is not universally true.
Let the radius of the circle be $r$.
The area of the circle is $A = \pi r^2$.
The circumference of the circle is $C = 2\pi r$.
Comparing the two,we look at the inequality $\pi r^2 > 2\pi r$.
Dividing both sides by $\pi r$ (assuming $r > 0$),we get $r > 2$.
Therefore,the area is greater than the circumference only when the radius $r > 2$.
If $0 < r < 2$,the circumference is greater than the area.
If $r = 2$,the numerical values are equal.
Thus,the statement is not universally true.
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