A rocket ascends vertically after being launched from a location that is midway between two ground-based tracking stations. When the rocket reaches an altitude of 4 kilometers, it is 5 kilometers from each of the tracking stations. Assuming that this is a locale where the terrain is flat, how far apart are the two tracking stations?
To solve this problem, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Let's assume that the two tracking stations are located at points A and B on the ground. The rocket is located at point R when it reaches an altitude of 4 kilometers.
We need to find the distance between the two tracking stations, which is the length of the segment AB. We can consider the triangle formed by points A, B, and R.
Since the rocket is 5 kilometers away from each tracking station when it is at an altitude of 4 kilometers, we can consider the triangle ARB as two right triangles, ARR' and BRR', where R' is the projection of R onto the ground.
In right triangle ARR', the rocket is located at point R, and the hypotenuse AR' is 5 kilometers (the distance from the rocket to one of the tracking stations) and the altitude RR' is 4 kilometers.
Using the Pythagorean theorem, we can calculate the length of the base side AR' as:
AR' = √(RR'^2 + AA'^2)
Substituting the values, we get:
AR' = √(4^2 + 5^2) = √(16 + 25) = √41
Similarly, in right triangle BRR', the length of side BR' is also √41.
The segment AB is the sum of the lengths of AR' and BR':
AB = AR' + BR' = √41 + √41 = 2√41 kilometers.
Therefore, the two tracking stations are approximately 2√41 kilometers apart.
Use the Pythagorean Theorem.
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