Answer
$z = 6.63\ m$
Work Step by Step
We know that the cable has 8 meters of cable, and that $x = 4\ m$ and $y= 2\ m$, then:
$r = [4i + 2j +zk]\ m$
To calculate the z-coordinate, we use the equation to calculate the magnitude and rearrange it:
$r = \sqrt{(i)^2 + (j)^2 + (z)^2}$
Substitute the know values and rearrange:
$8 = \sqrt{(4)^2 + (2)^2 + (z)^2} \Rightarrow 8^2 = (4)^2 + (2)^2 + (z)^2$
$64 = 16 + 4 + (z)^2 \Rightarrow 64 - 20 = (z)^2$
$z= \sqrt{44} = 6.63\ m$
