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?
I'm just stuck. Don't know exactly where to start.
Can't even draw a diagram?
Draw a vertical line. That's the rocket's path.
Label the bottom of the line B and place point T up somewhere and label BT with the number "4" the rocket's height.
Now draw a horizontal line through B, and at distance "x" mark points L and R for the left and right stations.
Now draw the two slanting lines LT and RT of length 5.
Now look. You have two right triangles with vertical leg 4 and hypotenuse 5.
Since x^2 + 4^2 = 5^2,
x = 3
so the distance LR between the stations is 3+3.
Got it?
To find the distance between the two tracking stations, we can utilize the concept of similar triangles. Let's denote the distance between the rocket and one of the tracking stations as 'x'.
From the given information, we know that when the rocket is at an altitude of 4 kilometers, it is 5 kilometers from each of the tracking stations. This forms a right-angled triangle with the rocket, one of the tracking stations, and the ground.
The altitude of the rocket forms a vertical line, creating two similar triangles. The smaller triangle is formed by the rocket, one of the tracking stations, and the altitude of the rocket, while the larger triangle is formed by the rocket, both tracking stations, and the altitude of the rocket.
In the smaller triangle, the altitude of the rocket is the vertical side, and the distance from the rocket to one of the tracking stations is the horizontal side ('x'). Since the rocket is 5 kilometers from each tracking station when the altitude is 4 kilometers, we can write:
(altitude of the rocket) / ('x') = 4 km / 5 km
simplifying this equation, we can solve for 'x':
('x') = 4 km / 5 km * (altitude of the rocket)
Plugging in the given value for the altitude of the rocket (4 kilometers), we can calculate 'x':
('x') = 4 km / 5 km * 4 km
('x') = 16 km / 5
('x') = 3.2 km
So, one of the tracking stations is 3.2 kilometers away from the rocket when the rocket is at an altitude of 4 kilometers.
Since the rocket is midway between the two tracking stations, the distance between the two tracking stations will be two times the distance between one of the tracking stations and the rocket:
2 * 3.2 km = 6.4 km
Therefore, the two tracking stations are approximately 6.4 kilometers apart.
if you just want confirmation of your hard work, what did you get?
If you got stuck, how far did you get? Did you draw a diagram? I think you'll see some more 3-4-5 triangles there.