A tree one hundred cubits high is distant from a well two hundred cubits; from this tree one monkey climbs dow the tree and goes to the well, but the other leaps in the air and descends by the hypotenuse from the high point of the leap, and both pass over an equal space. Find the height of the leap.

I assume you mean the monkey leaped straight up. Ignoring the fact that free-fall trajectories are parabolas, and not straight lines, we have

x = height of leap
300 = hypotenuse
(because the distance down the tree and over to the well is 300)

√(200^2 + (100+x)^2) = 300
x = 100(√5 - 1) = 123 cubits.

That's some leap!!

To solve this problem, we can first visualize the given information. Let's consider the scenario:

1. A tree is 100 cubits (units of length) high.
2. The well is 200 cubits away from the tree.
3. One monkey climbs down the tree and goes to the well.
4. The other monkey leaps in the air and descends by the hypotenuse.
5. Both monkeys pass over an equal space.

Let's proceed step by step to find the height of the leap:

Step 1: Draw a diagram
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Draw a right-angled triangle to represent the scenario. The vertical side of the triangle represents the height of the tree (100 cubits), and the horizontal side represents the distance between the tree and the well (200 cubits).

|
|
T |
|
|____________W

(T = Tree, W = Well)

Step 2: Apply Pythagoras' Theorem
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Using Pythagoras' Theorem, we know that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

According to our scenario, the height of the tree (T) is one of the sides, and the distance between the tree and the well (W) is another side. Let's assign variables to these sides:

T = 100 cubits (height of the tree)
W = 200 cubits (distance between the tree and the well)
H = height of the leap (hypotenuse)

Using Pythagoras' Theorem, we can write the equation as:

T^2 + W^2 = H^2

Now, substitute the given values:

(100)^2 + (200)^2 = H^2
10,000 + 40,000 = H^2
50,000 = H^2

Step 3: Find the height of the leap
===================================
To find the height of the leap (H), we can take the square root of both sides of the equation:

√50,000 = √H^2
H ≈ 223.61 cubits

Therefore, the height of the leap is approximately 223.61 cubits.