The Depth of the Lake: A Mathematical Solution

Problem Statement

A lake is surrounded by birds, and a lotus flower is seen 1/2 cubit above the water level. With the onset of wind, the lotus sinks 2 cubits away from its original position. How deep is the water in the lake?

Approach and Solution

While it might seem like the presence of birds and the wind movement provide additional context, we can solve this problem by focusing on the geometrical relationship between the lotus stalk, the depth of the water, and the resulting displacement.

Step 1: Initial Setup

Let's denote the depth of the lake as D.

When the lotus flower is observed 1/2 cubit above the water, it indicates that the length of the lotus stalk is D 1/2 cubits.

Step 2: Analyzing Movement Due to Wind

Upon the onset of wind, the lotus flower bends and sinks 2 cubits away from its original position, touching the surface of the water.

We can consider the path the lotus stalk takes as a right triangle with one leg being the depth of the water D, the other leg being 2 cubits, and the hypotenuse being the total length of the lotus stalk, which is D 1/2.

Step 3: Applying Pythagorean Theorem

The Pythagorean theorem states that for a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, we can set up the following equation:

D 1/22 D2 22

Simplifying the equation:

D2 D 1/4 D2 4

Subtract D2 from both sides:

D 1/4 4

Solving for D:

D 4 - 1/4

D 16/4 - 1/4

D 15/4

D 3.75

Therefore, the depth of the lake is 3.75 cubits.

Conclusion

Using the Pythagorean theorem, we were able to accurately determine the depth of the lake in the problem posed. This solution highlights the importance of utilizing geometric principles to solve real-world problems.