Answer
The distance between Karl's tent and Joe's tent is 28.2 meters.
Work Step by Step
Let west and south be positive directions.
Let $K$ be the vector from your tent to Karl's tent.
Let $J$ be the vector from your tent to Joe's tent.
Let $d$ be the vector from Karl's tent to Joe's tent.
$K+d = J$
$d = J - K$
We can find the west component $d_x$ of $d$.
$d_x = J_x - K_x$
$d_x = -21.0~cos(23.0^{\circ})~m - (-32.0)~cos(37.0^{\circ})~m$
$d_x = 6.23~m$ west
We can find the south component $d_y$ of $d$.
$d_y = J_y - K_y$
$d_y = 21.0~sin(23.0^{\circ})~m - (-32.0)~sin(37.0^{\circ})~m$
$d_y = 27.5~m$ south
We can use $d_x$ and $d_y$ to find the magnitude of $d$.
$d = \sqrt{(d_x)^2+(d_y)^2}$
$d = \sqrt{(6.23~m)^2+(27.5~m)^2}$
$d = 28.2~m$
The distance between Karl's tent and Joe's tent is 28.2 meters.
