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

make a sketch, on a straight line mark off points

X, Y, and Z, where Z is then end of the altitude SZ, where S is the satellite.
Angle X = 40°
angle SYZ=70°
XY = 24

in triangle XYS,
angle x=40, angle XYS = 110 , and angle XSY = 30°
by the sine law:
YS/sin40 = 24/sin30
YS = 30.8538

in right-angled triangle SYZ,
RZ/30.8538 = cos 70
RZ = 10.55

the height is appr. 10.6 km

Answer 2

thank you so much...

Answer 3

To find the altitude of the rocket, we can use trigonometry and the concept of similar triangles.

Let's denote the distance between the rocket and Station X as x, and the distance between the rocket and Station Y as y. The altitute of the rocket can be denoted as h.

From the given information, we can form two right triangles: Station X to the rocket and Station Y to the rocket. The angles of elevation, 40 degrees and 70 degrees, are the angles between the ground and the line of sight to the rocket.

In the triangle formed by Station X, the angle of elevation of 40 degrees, and the unknown height h, we have the following trigonometric relationship:

tan(40 degrees) = h / x

Similarly, in the triangle formed by Station Y, the angle of elevation of 70 degrees, and the unknown height h, we have the following trigonometric relationship:

tan(70 degrees) = h / y

Now, we can set up a relationship between x, y, and the distance between the two tracking stations (24 km):

x + y = 24

We can rearrange this equation to solve for y:

y = 24 - x

Now we can substitute this expression for y into the second trigonometric equation:

tan(70 degrees) = h / (24 - x)

Now we have a system of two equations with two variables:
1. tan(40 degrees) = h / x
2. tan(70 degrees) = h / (24 - x)

We can solve this system of equations to find the value of h.

First, rearrange equation 2 to solve for h:

h = (24 - x) * tan(70 degrees)

Substitute this expression for h into equation 1:

tan(40 degrees) = ((24 - x) * tan(70 degrees)) / x

Now, we can solve this equation for x. Rewrite it as:

x * tan(40 degrees) = (24 - x) * tan(70 degrees)

Expand both sides of the equation:

x * tan(40 degrees) = 24 * tan(70 degrees) - x * tan(70 degrees)

Rearrange the equation to solve for x:

x * (tan(40 degrees) + tan(70 degrees)) = 24 * tan(70 degrees)

x = (24 * tan(70 degrees)) / (tan(40 degrees) + tan(70 degrees))

Now, substitute this value of x back into the expression for h:

h = (24 - x) * tan(70 degrees)

h = (24 - [(24 * tan(70 degrees)) / (tan(40 degrees) + tan(70 degrees))]) * tan(70 degrees)

Using a calculator, we can evaluate this expression to obtain the value of h. The result will be the altitude of the rocket, correct to one decimal place.