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:

$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$

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:

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

$\left\{\begin{array}{l}x+y+z=10 \\ x^{2}+y^{2}+z^{2}=64 .\end{array}\right.$

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

![](https://cdn.mathpix.com/cropped/2024_05_06_424f80418dd801e5b28ag-40.jpg?height=657&width=625&top_left_y=1085&top_left_x=790)

Therefore, the desired curve is a circle with radius

$$
r=\sqrt{R^{2}-d^{2}}=\sqrt{64-\frac{100}{3}}=2 \sqrt{\frac{23}{3}},
$$

the length of which is $l=2 \pi r=4 \pi \sqrt{\frac{23}{3}}$ and is equal to the length of the cable being manufactured.

Answer. $4 \pi \sqrt{\frac{23}{3}}$.

## Variant-42. Solution