Question

Трасса, по которой бежит лыжник, состоит из четырёх этапов. Каждый этап (кроме первого) длиннее предыдущего на одну и ту же величину, и на каждый следующий этап (кроме первого) у лыжника ушло в два раза больше времени, чем на предыдущий. Известно также, что средняя скорость движения лыжника за время прохождения двух первых этапов совпадает со средней скоростью за время прохождения двух последних этапов. Во сколько раз последний этап длиннее первого?

Answer 1

Мда задачка конечно не втыкает в мозг

Answer 2

Problem Analysis

We are given that a skier is running on a track consisting of four stages. Each stage, except the first one, is longer than the previous stage by the same amount. Additionally, the skier takes twice as long to complete each subsequent stage compared to the previous stage. We are also told that the average speed of the skier during the first two stages is the same as the average speed during the last two stages. We need to determine how much longer the last stage is compared to the first stage.

Solution

Let's assume the length of the first stage is x units. Since each subsequent stage is longer than the previous stage by the same amount, the lengths of the stages can be represented as follows:

- First stage: x - Second stage: x + d - Third stage: x + 2d - Fourth stage: x + 3d

We are also given that the skier takes twice as long to complete each subsequent stage compared to the previous stage. Let's assume the time taken to complete the first stage is t units. Then the times taken to complete the subsequent stages can be represented as follows:

- First stage: t - Second stage: 2t - Third stage: 4t - Fourth stage: 8t

We are told that the average speed of the skier during the first two stages is the same as the average speed during the last two stages. The average speed can be calculated by dividing the distance traveled by the time taken. Therefore, we can set up the following equation:

Average speed for the first two stages = Average speed for the last two stages

The average speed for the first two stages can be calculated as follows:

Average speed for the first two stages = (x + (x + d)) / (t + 2t)

The average speed for the last two stages can be calculated as follows:

Average speed for the last two stages = ((x + 2d) + (x + 3d)) / (4t + 8t)

Setting these two equations equal to each other, we can solve for d, which represents the difference in length between each subsequent stage:

(x + (x + d)) / (t + 2t) = ((x + 2d) + (x + 3d)) / (4t + 8t)

Simplifying the equation:

(2x + d) / (3t) = (4x + 5d) / (12t)

Cross-multiplying:

(2x + d) * (12t) = (4x + 5d) * (3t)

Expanding and simplifying:

24xt + 12dt = 12xt + 15dt

Rearranging the equation:

12xt - 24xt = 15dt - 12dt

Simplifying further:

-12xt = 3dt

Dividing both sides by -3t:

4x = d

Therefore, the difference in length between each subsequent stage is d = 4x.

To find the ratio of the length of the last stage to the length of the first stage, we can substitute the value of d:

Length of the last stage = x + 3d = x + 3(4x) = 13x

Therefore, the last stage is 13 times longer than the first stage.

Answer

The last stage is 13 times longer than the first stage.