1. The points $A(-5,0), B(-5,7), C(-1,7)$, and $D(-1,4)$ are graphed, as shown. The line $L$ has equation $x=3$ and point $Q$ is the reflection of point $C$ in $L$.
(a) Point $P$ is the reflection of point $D$ in $L$. Point $R$ is the reflection of point $B$ in $L$. Point $S$ is the reflection of point $A$ in $L$. State the coordinates of $P, R$ and $S$.
(b) Calculate the perimeter of the figure $A B C D P Q R S$.
(c) Figure $A B C D P Q R S$ is rotated around the line $L$ to create a three-dimensional solid, as shown. This solid is a larger cylinder with a smaller cylinder removed. Determine the volume and the surface area of this solid. (Note: A cylinder with radius $r$ and height $h$ has volume $\pi r^{2} h$ and surface area $\left.2 \pi r^{2}+2 \pi r h.\right)$
1. (a) Points $D, B$ and $A$ are being reflected in a vertical line. Thus, the $y$-coordinate of each image point will be the same as the $y$-coordinate of the original point.
Point $D$ has $x$-coordinate -1 and is being reflected in the line $x=3$.
Point $D$ is thus $3-(-1)=4$ units to the left of the line $x=3$.
Therefore, its image after reflection will be 4 units to the right of the line $x=3$.
Thus, the coordinates of $P$ are $(3+4,4)=(7,4)$.
In a similar way, we reflect point $B(-5,7)$ in the line $x=3$ to obtain the point $R$ whose coordinates are $(3+8,7)=(11,7)$.
Finally, since the $x$-coordinate of $A$ is the same as the $x$-coordinate of $B$, then the $x$-coordinate of $S$ is the same as the $x$-coordinate of $R$.
Thus, the coordinates of $S$ are $(11,0)$.
(b) The perimeter of figure $A B C D P Q R S$ equals the sum of the lengths of its 8 sides. Each side is either vertical or horizontal. The length of each horizontal side can be found by subtracting $x$-coordinates; the length of each vertical side can be found by subtracting $y$-coordinates.
The perimeter equals $A B+B C+C D+D P+P Q+Q R+R S+S A$.
Given coordinates $A(-5,0)$ and $B(-5,7)$, we obtain $A B=7-0=7$.
Given coordinates $B(-5,7)$ and $C(-1,7)$, we obtain $B C=-1-(-5)=4$.
Similarly, $C D=3, D P=8, P Q=3, Q R=4, R S=7$, and $S A=16$.
Therefore, the perimeter of $A B C D P Q R S$ is $7+4+3+8+3+4+7+16=52$.
(Alternatively, the perimeter of the figure can be found by calculating the perimeter of the figure to the left of line $L$ and then doubling this total. This is because the figure is formed by reflecting the portion to the left of line $L$ to create the entire figure.)
(c) The larger cylinder has height equal to the length of $A B$, which is 7 , from part (b). The smaller cylinder has height equal to the length of $C D$, which is 3 from part (b). The radius of the larger cylinder equals the distance from $L$ to each of $B, R, S$, and $A$. In part (a), we saw that this distance is 8 .
The radius of the smaller cylinder equals the distance from $L$ to each of $C, Q, P$, and $D$. In part (a), we saw that this distance is 4 .
The volume of the given solid equals the volume of the larger cylinder minus the volume of the smaller cylinder.
Since the volume of a cylinder with radius $r$ and height $h$ is $\pi r^{2} h$, then the volume of the solid equals
$$
\pi\left(8^{2}\right)(7)-\pi\left(4^{2}\right)(3)=448 \pi-48 \pi=400 \pi
$$

The surface area of the given solid consists of the bottom and lateral faces of the larger cylinder (the outside), the bottom and lateral faces of the smaller cylinder (inside the hole), and the top face of the larger cylinder with the top face of the smaller cylinder removed.
Since the surface area of a cylinder with radius $r$ and height $h$ is $2 \pi r^{2}+2 \pi r h$, then the surface area of a cylinder with radius $r$ and height $h$ that has one end removed is $\left(2 \pi r^{2}+2 \pi r h\right)-\pi r^{2}=\pi r^{2}+2 \pi r h$.
Therefore, the surface area of the given solid is
$$
\left(\pi\left(8^{2}\right)+2 \pi(8)(7)\right)+\left(\pi\left(4^{2}\right)+2 \pi(4)(3)\right)+\left(\pi 8^{2}-\pi 4^{2}\right)=176 \pi+40 \pi+48 \pi=264 \pi
$$
(The term $\pi 8^{2}-\pi 4^{2}$ represents the area of the top circular face of the larger cylinder minus the area of the top circular face of the smaller cylinder.)
Thus, the volume of the given solid is $400 \pi$ and its surface area is $264 \pi$.