A rocket carrying a weather satellite is launched. As it moves through space, the rocket is tracked by two tracking stations located 24 km. apart, beneath the rocket. The 2 tracking stations both lie west of the launching pad. at a specific moment, the rocket's angle of elevation from Station X is 40 deg. while the rocket's angle of elevation from Station Y is 70 deg. Both tracking stations are west of the rocket at this moment. At this moment, what is the altitude of the rocket, correct to one decimal place?

Question

A rocket carrying a weather satellite is launched. As it moves through space, the rocket is tracked by two tracking stations located 24 km. apart, beneath the rocket. The 2 tracking stations both lie west of the launching pad. at a specific moment, the rocket's angle of elevation from Station X is 40 deg. while the rocket's angle of elevation from Station Y is 70 deg. Both tracking stations are west of the rocket at this moment. At this moment, what is the altitude of the rocket, correct to one decimal place?

Answer 1

To find the altitude of the rocket, we can use the concept of trigonometry and set up a right triangle with the rocket as the vertical side, the distance between the tracking stations as the base, and the angle of elevation as one of the angles.

Let's call the altitude of the rocket "h" and the distance between the tracking stations "d."

From the information given, we know that the angle of elevation from Station X is 40 degrees and the angle of elevation from Station Y is 70 degrees.

Now, we can use trigonometric functions to relate the angles and sides of the right triangle:

For Station X:
tan(40 degrees) = h / d

For Station Y:
tan(70 degrees) = h / (d + 24 km)

To find the altitude "h," we need to solve the two equations simultaneously. Rearranging the equations:

h = d * tan(40 degrees) --- (Equation 1)
h = (d + 24 km) * tan(70 degrees) --- (Equation 2)

Since both equations equal "h," we can set them equal to each other:

d * tan(40 degrees) = (d + 24 km) * tan(70 degrees)

Now, we can solve this equation for "d." Let's break it down step by step:

1. Convert the angles from degrees to radians:
tan(40 degrees) = tan(40 * pi / 180)
tan(70 degrees) = tan(70 * pi / 180)

2. Substitute the expressions back in:
d * tan(40 * pi / 180) = (d + 24 km) * tan(70 * pi / 180)

3. Solve for "d":
d = (24 km) * tan(70 * pi / 180) / (tan(70 * pi / 180) - tan(40 * pi / 180))

4. Substitute the value of "d" back into Equation 1 to find the altitude "h":
h = (24 km) * tan(40 * pi / 180) / (tan(70 * pi / 180) - tan(40 * pi / 180))

Calculating this expression will give you the altitude of the rocket at the specific moment, rounded to one decimal place.

yes

To find the altitude of the rocket, we can use the concept of trigonometry and set up a right triangle with the rocket as the vertical side, the distance between the tracking stations as the base, and the angle of elevation as one of the angles.