Task 6. (30 points) When manufacturing a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=-18 .
\end{array}\right.
$$

Find the length of the cable.
# Solution.

Transform the second equation of the system: square the first equation and subtract the second equation from it twice, we get:

$$
\begin{aligned}
& \left\{\begin{array}{l}
x+y+z=8, \\
(x+y+z)^{2}-2(x y+y z+x z)=8^{2}-2 \cdot(-18) .
\end{array}\right. \\
& \left\{\begin{array}{l}
x+y+z=8 \\
x^{2}+y^{2}+z^{2}=10 .
\end{array}\right.
\end{aligned}
$$

Then, the given curve is the intersection of a sphere centered at the origin with radius \( R=10 \) and the plane \( x+y+z=8 \), which is at a distance \( d=\frac{8}{\sqrt{3}} \) from the center of the sphere (Fig. 4).

![](https://cdn.mathpix.com/cropped/2024_05_06_424f80418dd801e5b28ag-45.jpg?height=657&width=626&top_left_y=1933&top_left_x=795)

Therefore, the desired curve is a circle with radius

$$
r=\sqrt{R^{2}-d^{2}}=\sqrt{100-\frac{64}{3}}=2 \sqrt{\frac{59}{3}},
$$

the length of which is \( l=2 \pi r=4 \pi \sqrt{\frac{59}{3}} \) and is equal to the length of the cable being manufactured.

Answer. \( 4 \pi \sqrt{\frac{59}{3}} \).