Functions of Several Variables — Question 8

Question 8 – Solution

Strategy Recognize the three-dimensional distance formula.

Interpretation $F$ is distance from $C=(1,-2,3)$, so its range is $\boxed{[0,\infty)}$.

Levels For $c>0$, $F=c$ is the sphere $$(x-1)^2+(y+2)^2+(z-3)^2=c^2.$$ For $c=0$ it is the single point $C$; for $c<0$ it is empty.

Requested surface The point $(4,2,3)$ is distance $\sqrt{3^2+4^2}=5$ from $C$, so its level surface is the sphere of radius $\boxed{5}$.