5. Maximum 15 points. The company "Intelligence, Inc" has developed a robot with artificial intelligence. To manufacture it, a special machine is required, which can produce 1 robot in 1 hour. The company owns a large number of such machines, but the created robot is so intelligent that it can produce an exact copy of itself in exactly 2 hours. "Intelligence, Inc" has received an urgent order for 3250 robots, which need to be manufactured in just 10 hours. The company's expenses for creating one robot (regardless of the time of its production and whether it was created by another robot or a special machine) are constant and amount to 100 monetary units, and the maintenance of one special machine during its operation to produce a robot costs the company 70 monetary units. "Intelligence, Inc" does not incur any other expenses for creating robots. What are the minimum costs the company will incur in fulfilling the order?
# Solution:

To minimize the company's costs, it is necessary to find the minimum number of machines that will allow the company to complete the order within the specified time frame.

Let $x$ be the number of machines. Then, in the first hour of operation, they will produce $x$ robots. These $x$ robots will start manufacturing new robots, and after 3 hours, there will be $2x$. Meanwhile, the machines continue their work and will produce a new batch of robots by the end of the 2nd hour, which will also start manufacturing their own kind. Thus, after 3 hours of the company's operation, there will be $2x + x$ robots. If we consider that after 9 hours from the start of the work on the order, the robots manufacturing their own kind will not have time to produce new ones by the deadline, then after 10 hours of operation, $62x$ robots will be produced.

The calculation of the number of robots is also presented in the table:

| 1 hour | 2 hour | 3 hour | 4 hour | 5 hour | 6 hour | 7 hour | 8 hour | 9 hour | 10 hour | Total |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathrm{x}$ | - | $2x + x$ | - | $6x + x$ | - | $14x + x$ | - | $30x + x$ | - | $60x + 2x$ |
| - | $\mathrm{x}$ | - | $2x + x$ | - | $6x + x$ | - | $14x + x$ | - | $30x + x$ |  |

It remains to find $x$ and round it up to the nearest whole number (otherwise, there won't be enough machines to produce the required number of robots within the specified time).

$$
x = \frac{3250}{62} \approx 52.42
$$

Thus, the company's costs for fulfilling the order will be $100 * 3250 + 70 * 53 = 328710$ monetary units.

## Grading Scheme:

Full correct solution 15 points.

Correctly found the number of robots - 10 points out of 15.

Correctly found the company's expenses for fulfilling the order - 5 points out of 15.

The number of robots is found incorrectly, but there is a correct logic for finding the costs - 2 points out of 15.

There is progress in finding the number of robots, but there are minor logical errors and the number of robots is found incorrectly - 5 points out of 15.