Hello, aligator0207!

Here's part (a).

A steel pipe is being carried down a hallway 9 ft. wide.

At the end of the hall there is a right-angled turn into a narrower hallway 6 ft. wide.

a.) Show that the length of the pipe in the figure is modeled by the function:

. . . . . $\displaystyle L(\theta) \:=\: 9\csc\theta + 6\sec\theta$ Code:

C
*-------------------------------o------
| * θ :
| * :6
| B * :
| o---------------o------
| * θ | E
| θ * |
| * |
A o - - - - - - - o D
| 9 |
| |

The pipe is: .$\displaystyle L \:=\:AC \:=\:AB+BC$

$\displaystyle AD = 9,\;CE = 6$

Note that: .$\displaystyle \angle ABD \,=\,\angle BCE \,=\,\theta$

In right triangle $\displaystyle BDA\!:\;\;\csc\theta \:=\:\frac{AB}{9} \quad\Rightarrow\quad AB \:=\:9\csc\theta$

In right triangle $\displaystyle CEB\!:\;\;\sec\theta \:=\:\frac{BC}{6} \quad\Rightarrow\quad BC \:=\:6\sec\theta$

Therefore: .$\displaystyle L \;=\;9\csc\theta + 6\sec\theta$