Problem 48 - Algebra and Trigonometry

Question

Gateway Arch A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of \(35,000\) 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) 93,162.91 ft (b) 86,980.14 ft

Step 1: Understanding the Problem

There are two distance calculations we need: the distance between the plane and the Gateway Arch, and the horizontal ground distance between the point directly below the plane and the Gateway Arch. Given: the plane's elevation (height) is 35,000 ft and the angle of depression is 22 degrees.

Step 2: Trigonometric Relationship

When the angle of depression is given, it is equal to the angle of elevation from the corresponding point on the ground. This creates a right triangle where the angle of elevation is 22°, the height (opposite side) is 35,000 ft, and we need to find the adjacent side (horizontal distance to point below the plane) and the hypotenuse (distance from the plane to the arch).

Step 3: Calculate Horizontal Ground Distance

To find the horizontal ground distance (adjacent side), use the tangent function: \[\tan(22°) = \frac{\text{opposite}}{\text{adjacent}}\ = \frac{35,000}{d_{adj}}\]Solving for \(d_{adj}\) gives:\[d_{adj} = \frac{35,000}{\tan(22°)}\]Using a calculator, \(d_{adj} ≈ 86,980.14\) ft.

Step 4: Calculate Distance from Plane to Arch

To find the distance from the plane to the arch (hypotenuse), use the cosine function:\[\cos(22°) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{d_{adj}}{d_{hyp}}\]Solving for \(d_{hyp}\) gives:\[d_{hyp} = \frac{86,980.14}{\cos(22°)}\]Using a calculator, \(d_{hyp} ≈ 93,162.91\) ft.

Key Concepts

Angle of Depression

Imagine you are seated in a plane looking down at the ground. The angle that your line of sight makes with a straight line from the horizontal plane (parallel to the ground) is known as the angle of depression. In trigonometry, this angle has a special property: it is equal to the angle of elevation from the point on the ground directly under your sightline back up to the plane.
This fact helps us solve problems involving heights and distances, like in this exercise with the plane and the Gateway Arch.

• Angle of depression = Angle of elevation
• Forms a right triangle with the line of sight and ground

Understanding this concept allows you to effectively find horizontal distances or heights in problems involving trigonometry.

Right Triangle

A right triangle is a triangle where one of the angles is exactly 90 degrees. In such triangles, the Pythagorean theorem holds, and most importantly for trigonometry, the basic functions of sine, cosine, and tangent are often used based on its angles and sides.
In the context of our problem, a right triangle is formed by:

• The height of the plane above the ground (the opposite side)
• The horizontal distance on the ground (the adjacent side)
• The line of sight (the hypotenuse, longest side)

These components fit perfectly into trigonometric functions that help us solve for missing distances, leveraging the relationships between angles and side lengths.

Tangent Function

The tangent function is one of the primary trigonometric functions used in right triangles. It relates the opposite side of the triangle to the adjacent side. This concept is key in solving our exercise to find the ground distance from directly below the plane to the arch. The tangent of an angle \( \theta \) in a right triangle is defined as: \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \] In our Gateway Arch example:

• The opposite side is the height of the plane above the ground, 35,000 ft.
• The adjacent side is the horizontal distance on the ground, \(d_{adj}\).

By rearranging the formula to solve for \(d_{adj}\), we get that equation we used, \(d_{adj} = \frac{35,000}{\tan(22°)}\). This calculation helps us find the much-needed horizontal distance.

Cosine Function

The cosine function is another fundamental trigonometric function used with right triangles. It relates the adjacent side of a right triangle to the hypotenuse. In our problem, knowing the distance directly below the plane (adjacent side) allowed us to solve for the distance from the plane to the Gateway Arch using the cosine function. The cosine of an angle \( \theta \) is given by:\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \] The adjacent side here is the already found ground distance \(d_{adj}\) of approximately 86,980.14 ft, and the hypotenuse is the distance we need. By rearranging and solving the equation \(d_{hyp} = \frac{d_{adj}}{\cos(22°)}\), we effectively determine the straight-line distance from the plane to the arch, making the cosine function a potent tool in trigonometric applications.