Question

1. for problems 6 and 7, sketch a polygon inside the circle and a polygon that contains the circle, to determine a range for the area of the circle.

Explanation:

Step1: Identify circle parameters

The circle has center $(4,4)$ and radius $3$ (distance from center to leftmost point $x=1$, or rightmost $x=7$, top $y=1$, bottom $y=7$).

Step2: Inscribe square in circle

The inscribed square has vertices at $(4\pm\frac{3\sqrt{2}}{2}, 4\pm\frac{3\sqrt{2}}{2})$. Its side length is $3\sqrt{2}$, so area is $(3\sqrt{2})^2 = 18$.

Step3: Circumscribe square around circle

The circumscribed square has vertices at $(1,1), (7,1), (7,7), (1,7)$. Side length is $6$, area is $6^2=36$.

Step4: Define area range

The circle's area lies between the inscribed and circumscribed polygon areas.

Answer:

The area of the circle is between 18 and 36, so the range is $18 < \text{Area of circle} < 36$.
(Note: A square inscribed inside the circle and a square containing the circle are valid polygons for this problem.)