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:

$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=14
\end{array}\right.
$$

Find the length of the cable.
# Solution.

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

$\left\{\begin{array}{l}x+y+z=8, \\ (x+y+z)^{2}-2(x y+y z+x z)=8^{2}-2 \cdot 14 .\end{array}\right.$

$\left\{\begin{array}{l}x+y+z=8 \\ x^{2}+y^{2}+z^{2}=36 .\end{array}\right.$

Then, the given curve is the intersection of a sphere with center at the origin and radius $R=6$ and the plane $x+y+z=8$, which passes at a distance $d=\frac{8}{\sqrt{3}}$, which is less than 6, from the center of the sphere (Fig. 4).

![](https://cdn.mathpix.com/cropped/2024_05_06_424f80418dd801e5b28ag-34.jpg?height=646&width=622&top_left_y=1887&top_left_x=794)

Fig. 4

Therefore, the desired curve is a circle with radius

$$
r=\sqrt{R^{2}-d^{2}}=\sqrt{36-\frac{64}{3}}=2 \sqrt{\frac{11}{3}}
$$

the length of which is $l=2 \pi r=4 \pi \sqrt{\frac{11}{3}}$ and is equal to the length of the cable being manufactured.

Answer. $4 \pi \sqrt{\frac{11}{3}}$.

## Mathematics. 10-11 grade.

## Variant-41