Problem 48 - Precalculus Mathematics for Calculus

Question

A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of \(35,000 \mathrm{ft}\). The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is \(22^{\circ} .\) (a) What is the distance between the plane and the arch? (b) What is the distance between a point on the ground directly below the plane and the arch?

Step-by-Step Solution

Answer

(a) 93,796 ft (b) 86,412 ft

Step 1: Understand the problem and use the angle of depression

We know the plane is at an elevation of 35,000 ft. The angle of depression to a point on the ground below the Gateway Arch is given as \(22^{\circ}\). This means the line of sight from the plane to the point on the ground forms an angle of \(22^{\circ}\) with the horizontal line from the plane.

Step 2: Set up the right triangle

Consider a right triangle where: - The vertical leg is the height from which the plane is flying, i.e., 35,000 ft. - The horizontal leg is the distance between a point directly below the plane to the point on the ground below the Gateway Arch. - The hypotenuse is the direct line of sight distance from the plane to the point on the ground below the Gateway Arch.

Step 3: Calculate the horizontal distance using trigonometry for part (b)

To find the horizontal distance between the point directly below the plane and the arch, we use the tangent of the angle of depression. Using \( \tan(22^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} \), we set:\[ \tan(22^{\circ}) = \frac{35000}{x} \] Solve for \(x\):\[ x = \frac{35000}{\tan(22^{\circ})} \approx 86,412 \text{ ft} \]

Step 4: Calculate the hypotenuse for part (a)

To find the distance from the plane to the Gateway Arch (hypotenuse of the triangle), use the sine of the angle of depression. From right-angle triangle trigonometry:\[ \sin(22^{\circ}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{35000}{c} \]Solve for \(c\):\[ c = \frac{35000}{\sin(22^{\circ})} \approx 93,796 \text{ ft} \]

Key Concepts

Angle of Depression

In trigonometry, the angle of depression is the angle formed when looking down from an observer's eye level to an object situated below. This angle is always measured downwards from a horizontal line of sight. In the exercise described, the angle of depression is the angle between the plane's line of sight directly towards a point on the ground below and the horizontal line from the plane's position. For the pilot flying the plane at an altitude of 35,000 ft, this angle is given as \(22^{\circ}\). When working with such problems, it's crucial to remember that the angle of depression from the plane is equal in measure to the angle of elevation from the ground up to the plane, due to alternate interior angles in parallel lines. Therefore, understanding this concept helps solve for distances in vertical scenarios, using the relationships made available in right triangles.

Right Triangle

A right triangle is a triangle where one angle measures exactly 90 degrees. This type of triangle is foundational in trigonometry and helps students understand relationships between different sides and angles. In the problem, the situation with the plane creates a right triangle where:

• The vertical side (or leg) is the altitude of the plane, which is 35,000 ft.
• The horizontal side is the distance from a point directly below the plane to the ground point under the Gateway Arch—this is part of what needs to be calculated.
• The hypotenuse represents the direct line of sight from the plane to the point on the ground below the Gateway Arch.

Understanding the layout of these components helps use trigonometric ratios effectively to find unknown distances.

Trigonometric Ratios

Trigonometric ratios involve the relationships between angles and sides of right triangles. These include sine (sin), cosine (cos), and tangent (tan). They are the cornerstone of solving right triangle problems in trigonometry. In this exercise:

• Tangent ratio is used to find the horizontal distance (base of the triangle): \[ \tan(22^{\circ}) = \frac{\text{height}}{\text{base}} = \frac{35000}{x} \] Solving for \(x\) gives the distance directly below the plane to a point on the ground below the Gateway Arch as approximately 86,412 ft.
• To find the hypotenuse (direct line of sight), the sine ratio helps: \[ \sin(22^{\circ}) = \frac{35000}{c} \] Solving this, the hypotenuse or direct sight line equals about 93,796 ft.

Using these ratios allows the calculation of unknown sides when one side and one non-right angle of a right triangle are known, showcasing the power and utility of trigonometric concepts in real-world problems.