A fox and an eagle lived at the top of a cliff of height 6m,
whose base was at a distance of y metres from a point A on the
ground. The fox descends the cliff and went straight to the point
A. The eagle flew vertically up to a height x metres and then flew
in a straight line to a point A, the distance traveled by each
being the same. Find the value of x if y is three times the x.
2.72m
To solve this problem, let's break it down step by step.
1. Let's start by visualizing the scenario. We have a cliff of height 6m, and the base of the cliff is at a distance of y meters from point A on the ground. The fox descends the cliff and goes straight to point A. The eagle, on the other hand, flies vertically up to a height x meters, and then flies in a straight line to point A.
2. We need to find the value of x, given that y is three times x.
3. Let's use the concept of similar triangles. Since the distance traveled by each animal is the same, we can set up a proportion for the two triangles formed between the cliff and the distance traveled by each animal.
Let's call the distance traveled by the fox as d1, and the distance traveled by the eagle as d2.
We have the following proportion: d1 / 6 = d2 / x
Since y is three times x, we can substitute y with 3x: d1 / 6 = d2 / (3x)
4. Now, we need to find the relationship between d1 and d2. Since the fox descends the cliff and goes straight to point A, it forms a right-angled triangle with the cliff, where the cliff's height is the vertical side and d1 is the hypotenuse.
Similarly, the eagle flies vertically up to a height x and then travels a horizontal distance to reach point A. This forms a right-angled triangle where the height x is the vertical side and d2 is the hypotenuse.
Using the Pythagorean theorem, we have:
For the fox: d1^2 = 6^2 + y^2
For the eagle: d2^2 = x^2 + y^2
5. We can substitute y with 3x in the equation for d2^2: d2^2 = x^2 + (3x)^2
Simplifying this equation: d2^2 = x^2 + 9x^2
d2^2 = 10x^2
6. Now, let's substitute d1 and d2 in the proportion we set up earlier: (d1 / 6) = (d2 / (3x))
Substituting the values we found: (sqrt(6^2 + y^2) / 6) = (sqrt(10x^2) / (3x))
Since the distance traveled cannot be negative, we will only consider the positive square root.
Simplifying this equation: sqrt(6^2 + y^2) / 6 = sqrt(10) / 3
7. Now, let's solve for y: sqrt(6^2 + y^2) = (sqrt(10) / 3) * 6
Simplifying this equation: sqrt(36 + y^2) = 2 * sqrt(10)
Squaring both sides of the equation to eliminate the square root: 36 + y^2 = 4 * 10
Simplifying this equation: y^2 = 40 - 36
y^2 = 4
Taking the square root of both sides: y = 2
8. Since y is three times x, we have: x = y/3 = 2/3
Therefore, the value of x is 2/3 meters.
In conclusion, the value of x is 2/3 meters.