Hello, dgenerationx2!

2. Two boats start at the dock.

The 1st sails 4 miles due west, then turns 46° south and sails 1.8 miles.

The 2nd sails 4 miles due east, then turns 58° north and sails 1.8 miles.

Which boat is farther from the dock? Code:

* B
* /
* /1.8
* /
P 4 * 122°/58°
W - - - * - - - - o - - - - * - - - E
46°/134° * D 4 Q
/ *
1.8 / *
/ *
A *

The 1st boat starts at $\displaystyle D$, sails 4 miles west to $\displaystyle P$,

. . then turns 46° south and sails 1.8 miles to $\displaystyle A.$

$\displaystyle \angle WPA = 46^o \quad\Rightarrow\quad \angle DPA = 134^o$

The 2nd boat starts at D, sails 4 miles east to Q,

. . then turns 58° north and sails 1.8 miles to $\displaystyle B.$

$\displaystyle \angle BQE = 58^o \quad\Rightarrow\quad \angle DQB = 122^o$

We will use the Law of Cosines . . .

In $\displaystyle \Delta APD\!:\;\;AD^2\:=\:1.8^2 + 4^2 - 2(1.8)(4)\cos134^o \:=\:29.24308053$

. . Hence: .$\displaystyle D \:\approx\:5.4$ miles.

In $\displaystyle \Delta DQB\!:\;\;DB^2 \:=\:4^2 + 1.8^2 - 2(4)(1.8)\cos122^o \:=\:26.8708324$

. . Hence: .$\displaystyle DB \:\approx\:5.2$ miles.

Therefore, the first boat is farther from the dock.