Question

У Васи есть доступ к Интернет по высокоскоростному одностороннему радиоканалу, обеспечивающему скорость получения им информации 2^17 бит в секунду. У Пети нет скоростного доступа в Интернет, но есть возможность получать информацию от Васи по низкоскоростному телефонному каналу со средней скоростью 2^15 бит в секунду. Петя договорился с Васей, что тот будет скачивать для него данные объемом 4 Мбайта по высокоскоростному каналу и ретранслировать их Пете по низкоскоростному каналу. Компьютер Васи может начать ретрансляцию данных не раньше, чем им будут получены первые 512 Кбайт этих данных. Каков минимально возможный промежуток времени (в секундах), с момента начала скачивания Васей данных, до полного их получения Петей? В ответе укажите только число, слово «секунд» или букву «с» добавлять не нужно.

Answer 1

Problem Analysis

Vasya has high-speed internet access with a speed of 2^17 bits per second, while Petya has low-speed internet access with a speed of 2^15 bits per second. Vasya has agreed to download data of size 4 megabytes (4 MB) using his high-speed connection and then transmit it to Petya over the low-speed connection. We need to determine the minimum possible time it takes for Petya to receive the complete data from the moment Vasya starts downloading.

Solution

To calculate the minimum possible time for Petya to receive the complete data, we need to consider the following:

1. Vasya's high-speed connection speed: 2^17 bits per second. 2. Petya's low-speed connection speed: 2^15 bits per second. 3. Data size to be downloaded by Vasya: 4 megabytes (4 MB).

Let's calculate the time it takes for Vasya to download the data and the time it takes for Petya to receive the complete data.

1. Time taken by Vasya to download the data: - Data size: 4 megabytes (4 MB) = 4 * 8 * 2^20 bits. - Vasya's connection speed: 2^17 bits per second. - Time taken by Vasya to download the data = (Data size) / (Vasya's connection speed).

2. Time taken by Petya to receive the complete data: - Petya's connection speed: 2^15 bits per second. - Petya can start receiving the data only after the first 512 kilobytes (512 KB) have been downloaded by Vasya. - Remaining data size to be transmitted to Petya: (Data size) - (512 kilobytes). - Time taken by Petya to receive the complete data = (Remaining data size) / (Petya's connection speed).

To find the minimum possible time, we need to consider the maximum time taken between Vasya's download time and Petya's reception time.

Calculation

Let's calculate the minimum possible time for Petya to receive the complete data.

1. Time taken by Vasya to download the data: - Data size = 4 * 8 * 2^20 bits. - Vasya's connection speed = 2^17 bits per second. - Time taken by Vasya to download the data = (4 * 8 * 2^20) / (2^17) seconds.

2. Time taken by Petya to receive the complete data: - Petya's connection speed = 2^15 bits per second. - Remaining data size to be transmitted to Petya = (4 * 8 * 2^20) - (512 * 8 * 2^10) bits. - Time taken by Petya to receive the complete data = (Remaining data size) / (2^15) seconds.

The minimum possible time is the maximum of the above two times.

Answer

The minimum possible time for Petya to receive the complete data, from the moment Vasya starts downloading, is (4 * 8 * 2^20) / (2^17) seconds.

Note: The above calculation assumes that there is no additional delay or overhead in the transmission process.