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1. Rotation center of a complex map ●●○○○ 20 ptsThe function $f(z) = \frac{(-1 + i \sqrt{3})\, z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$, writing $c = x + yi$ with $x$ and $y$ real. Round $x$ to two decimal places.
2. Area swept by a bent segment ●●●○○ 50 ptsPlot points $A, B, C$ at coordinates $(0,0)$, $(0,1)$ and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $AB$ and $BC$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise $45^\circ$ about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise $45^\circ$ about point $A$. Find $\frac{X_{1}+X_{2}}{2}$. Round to two decimal places.
3. Area under a rotating rectangle's path ●●●○○ 50 ptsA $2 \times 3$ rectangle has vertices $(0,0)$, $(2,0)$, $(0,3)$, $(2,3)$. Rotate it $90^{\circ}$ clockwise about $(2,0)$, then $90^{\circ}$ clockwise about $(5,0)$, then $90^{\circ}$ clockwise about $(7,0)$, and finally $90^{\circ}$ clockwise about $(10,0)$, so that the side originally on the $x$-axis is back on the $x$-axis. A point of the rectangle initially at $(1,1)$ traces a curve during these rotations. Find the area of the region below this curve and above the $x$-axis. Round to two decimal places.
4. Rotating a line to equal intercepts ●●●○○ 50 ptsThe line $y=10-2x$ is rotated anticlockwise about the origin through an angle $\theta$, where $0 \le \theta \le 90$ (in degrees). The image has $x$-intercept $(p,0)$ and $y$-intercept $(0,q)$. Determine $\theta$ such that $p=q$. Round to two decimal places.