Problem 31 - Algebra and Trigonometry

Question

Tracking a Satellite The path of a satellite orbiting the earth causes it to pass directly over two tracking stations \(A\) and \(B,\) which are 50 \(\mathrm{mi}\) apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(87.0^{\circ}\) and \(84.2^{\circ},\) respectively. (a) How far is the satellite from station \(A\) ? (b) How high is the satellite above the ground?

Step-by-Step Solution

Answer

(a) 323.6 miles (b) 323.3 miles above the ground

Step 1: Identify Known Values

The distance between tracking stations \(A\) and \(B\) is 50 miles. The angle of elevation at \(A\) is \(87.0^{\circ}\) and at \(B\) is \(84.2^{\circ}\).

Step 2: Draw a Diagram

Visualize the problem by sketching a triangle where the satellite is at the point where angles \(87.0^{\circ}\) and \(84.2^{\circ}\) meet, and the horizontal distance between \(A\) and \(B\) makes the base.

Step 3: Use the Law of Sines

Label the satellite as point \(C\). In triangle \(ACB\), use the Law of Sines: \[ \frac{AC}{\sin B} = \frac{BC}{\sin A} = \frac{AB}{\sin C} \]with \(\angle ACB = 180^{\circ} - 87^{\circ} - 84.2^{\circ}\) to find \(AC\).

Step 4: Calculate the Angle at C

Calculate \(\angle ACB = 8.8^{\circ}\) since \[180^{\circ} - (87.0^{\circ} + 84.2^{\circ}) = 8.8^{\circ} \]

Step 5: Substitute Values into Law of Sines

Using the Law of Sines, \[ \frac{AC}{\sin 84.2^{\circ}} = \frac{50}{\sin 8.8^{\circ}} \]

Step 6: Calculate Distance from Station A

Rearrange to find \(AC\):\[AC = \frac{50 \times \sin 84.2^{\circ}}{\sin 8.8^{\circ}} \approx 323.6 \text{ miles} \] Thus, the satellite is approximately 323.6 miles from Station \(A\).

Step 7: Calculate Satellite's Height Above Ground

Now use right triangle trigonometry with the angle at \(A\):\[ \text{Height} = AC \times \sin 87^{\circ} \approx 323.6 \text{ miles} \times \sin 87^{\circ} \approx 323.3 \text{ miles} \] So, the satellite is approximately 323.3 miles above the ground.

Key Concepts

Law of Sines

In trigonometry, the Law of Sines is a helpful formula to solve for unknown parts of a triangle. This law is especially useful in non-right triangles where traditional trigonometric methods don't apply directly. The Law of Sines states that in any triangle with angles A, B, and C, and sides opposite to these angles as a, b, and c respectively, the relationship can be expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This equation suggests a proportional relationship between the lengths of the sides and the sines of the opposite angles.

• Use it to find unknown side lengths if two angles and one side are known.
• Excellent for applying to problems involving oblique (non-right) triangles.

In our satellite problem, the Law of Sines was essential to find how far the satellite is from Station A. By using the known side and angles, we could calculate side AC, which is the distance we need.
Understanding the basic application of this law helps in correctly setting up and solving the equations for similar exercises.

Angles of Elevation

Angles of elevation are angles measured from the horizontal upward to an object. They are commonly used in real-world scenarios like finding the height of a tower, a tree, or in this case, tracking a satellite.

• An angle of elevation forms a right triangle with the object, observer's eye level, and the ground as parts of the triangle.
• This concept is critical in determining the position and altitude of objects above the observer's horizontal line.

In our scenario, the angles of elevation from station A and B to the satellite were given as 87° and 84.2°, respectively.
These angles are the key to identifying the necessary triangle components to apply the Law of Sines and further calculations. By understanding these angles, students can visualize and plan the correct approach to solving similar spatial problems.

Triangles

Triangles are fundamental shapes in geometry that consist of three sides and three angles. They are classified by either the angles or the lengths of their sides.

• Right triangles have one angle equal to 90 degrees.
• Acute triangles have all angles less than 90 degrees.
• Obtuse triangles have one angle more than 90 degrees.

In the exercise, we dealt with an obtuse triangle made by the points formed by the satellite and the two tracking stations, A and B.
Visualizing this triangle helps in framing the problem, setting the known values, and choosing the right trigonometric rule to apply. Each angle and side serves specific roles and influences which mathematical formulas are suitable.

Distance Calculation

Distance calculation in trigonometry involves combining angles and sides of triangles to find unknown lengths. It leverages trigonometric relationships and properties, mostly when tackling oblique triangles.

• The Law of Sines is a primary method for non-right triangles.
• Trigonometric functions like sine, cosine, or tangent help relate angles with sides.

In our step-by-step solution, determining the distance of the satellite from the tracking station involved rearranging the formula derived from the Law of Sines. By solving for AC, the student found the distance using known angles and the baseline of the tracking stations.
By understanding these calculation methods, a student can extend these principles to find not just distances, but also heights and other unknown values in geometrical contexts.