Question

В сказочном парке был пруд, в котором росла лилия. Каждый день число листьев удваивалось, и на двадцатый день вся поверхность пруда закрывалась листьями. Какая часть пруда оставалась свободной от листьев через 18 дней?

Answer 1

На 18 день была свободна 1/4 пруда

Answer 2

Problem Analysis

In this problem, we are given a pond in a fairy park that has a lily growing in it. Every day, the number of lily leaves doubles, and on the twentieth day, the entire surface of the pond is covered with leaves. We need to determine what fraction of the pond's surface remains free of leaves after eighteen days.

Solution

To solve this problem, we can work backwards from the twentieth day and calculate the fraction of the pond's surface covered with leaves on each day. Then, we can subtract this fraction from 1 to find the fraction of the pond's surface that remains free of leaves.

Let's calculate the fraction of the pond's surface covered with leaves on each day:

- On the twentieth day, the entire surface of the pond is covered with leaves. Therefore, the fraction of the pond's surface covered with leaves on the twentieth day is 1.

- On the nineteenth day, the number of leaves is half of what it was on the twentieth day. Therefore, the fraction of the pond's surface covered with leaves on the nineteenth day is 1/2.

- On the eighteenth day, the number of leaves is half of what it was on the nineteenth day. Therefore, the fraction of the pond's surface covered with leaves on the eighteenth day is 1/4.

We can observe a pattern here: the fraction of the pond's surface covered with leaves on each day is halved compared to the previous day. Therefore, we can conclude that the fraction of the pond's surface covered with leaves on the n-th day is given by 1/2^(20-n).

Now, let's calculate the fraction of the pond's surface that remains free of leaves after eighteen days:

- On the eighteenth day, the fraction of the pond's surface covered with leaves is 1/4. Therefore, the fraction of the pond's surface that remains free of leaves is 1 - 1/4 = 3/4.

Answer

After eighteen days, 3/4 of the pond's surface remains free of leaves.

Explanation

On each day, the number of leaves doubles, and the fraction of the pond's surface covered with leaves is halved compared to the previous day. Therefore, after eighteen days, the fraction of the pond's surface that remains free of leaves is 3/4.