A small buoy is tied to a rope and moves in a lake, with all distances in metres…

A small buoy is tied to a rope and moves in a lake, with all distances in metres. A coordinate system has its origin at the anchor. The buoy is at the point $(t,\ t + 1,\ t + 2)$ when a control knob is set to the number $t$. The operator wants the buoy to be exactly $5\sqrt{2}\ \text{m}$ from the anchor. This happens for two values of $t$. Write the square of the distance from the origin as a quadratic expression in $t$ and set it equal to $50$. Solve for $t$. Give the positive value of $t$ and the negative value of $t$. Then say in which octant the buoy sits for the negative value, as a number from 1 to 8.