Question

PLEASE HELP ASAP!!!! 100% CORRECT ANSWERS ONLY PLEASE!!!! I CANNOT RETAKE THIS

Answer 1

The first statement is true. Reasoning below.

= = =

We want to find the area of a fixed circle, so we can throw out the last option. If $r$ changes at all, then so does the area of the circle.

For $s$ to increase would require using a circumscribed polygon with less sides. Again, the circle is fixed, so only a certain length $s$ can fit inside the circle. This eliminates the third option.

Note that if we use a regular hexagon, then $s=r$ automatically, because the component triangles that make up the hexagon are equilateral. Increasing $h$ would require that we use a polygon with more sides, which would simultaneously make $s$ stray away from $r$. In other words, if $h$ increases, then $s$ decreases, so we can never eventually have $s\to r$ ($r$ is fixed).

That leaves the first option. Indeed, as $n$ increases, we get a polygon that looks increasingly rounder and more like a perfect circle. At the same time, that means $h$ gets larger, but would be bounded above by the circle's perimeter. So as $h$ increases indefinitely, it will eventually "be equal" (in the limit sense) to $r$, so that $rh\to r^2$.