7. (10 points) An industrial robot travels from point $A$ to point $B$ according to a pre-determined algorithm. The diagram shows a part of its repeating trajectory. Determine how many times faster it would reach from point $A$ to point $B$ if it moved in a straight line at three times the speed?

![](https://cdn.mathpix.com/cropped/2024_05_06_0588747f110ff428b268g-4.jpg?height=306&width=724&top_left_y=190&top_left_x=766)
Answer: 9 times.

Solution. The time spent by the robot moving along the given trajectory:

$t_{1}=N \frac{2 a+3 a+a+3 a}{v}=9 \frac{N a}{v}$.

The time spent by the robot moving in a straight line:

$t_{2}=N \frac{2 a+a}{3 v}=\frac{N a}{v}$.

We obtain that the robot will arrive $\frac{t_{1}}{t_{2}}=9$ times faster.