Question

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 90 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.0° and 84.2°, respectively. (Round your answers to the nearest mile.)
(a) How far is the satellite from station A? mi
(b) How high is the satellite above the ground? mi

Answer 1

Answer:

A) 585 miles

B) 584 miles

Step-by-step explanation:

We are told that the satellite pass directly over two tracking stations A and B, which are 90 mi apart.

This means that we can look at this as forming a kind of triangle where Tracking stations A & B will be

opposite ends of the base of the triangle while the satellite position will be the apex of the triangle.

We are given the angles of elevation at A and B to be 87.0° and 84.2° respectively.

Now, if we depict the satellite position as C. Then we have;

Angle at C = 180 - (87 + 84.2)

Angle at C = 8.8°

I've attached an image depicting this triangle.

A) To find the distance of the satellite from Station A, from the image attached, the distance will be AC.

Since AB is 90 mi.

Thus, using sine rule, we have;

AB/sin C = AC/sinB

Plugging in the relevant values;

90/(Sin 8.8) = AC/(sin 84.2)

AC = (90 × (sin 84.2))/(Sin 8.8)

AC = 585.27

To the nearest mile gives;

AC = 585 miles

B) To find out how high the satellite is above the ground, is the perpendicular distance from point C to line AB

Let's call the point at which the line touches AB as D.

It means angle D is 90° and it means triangle ADC it's a right angled triangle where AC is the hypotenuse = 585 miles.

Thus, using Trigonometric ratios we can find CD

Sin(87) = CD/585

CD = (Sin 87) x 585

CD = 584.198 miles

Approximating to the nearest mile gives;

CD = 584 miles