Problem 32 - College Trigonometry

Question

Skippy and Sally decide to hunt UFOs. One night, they position themselves 2 miles apart on an abandoned stretch of desert runway. An hour into their investigation, Skippy spies a UFO hovering over a spot on the runway directly between him and Sally. He records the angle of inclination from the ground to the craft to be \(75^{\circ}\) and radios Sally immediately to find the angle of inclination from her position to the craft is \(50^{\circ} .\) How high off the ground is the UFO at this point? Round your answer to the nearest foot. (Recall: 1 mile is 5280 feet.)

Step-by-Step Solution

Answer

The UFO is approximately 2473 feet off the ground.

Step 1: Convert Miles to Feet

Given that 1 mile is 5280 feet, we first convert the 2 miles between Skippy and Sally into feet. Thus, 2 miles = 2 x 5280 feet = 10560 feet.

Step 2: Set Up the Right Triangle for Skippy's Side

Since Skippy sees the UFO at a 75-degree angle, we can form a right triangle with the horizontal distance as one leg, the line of sight to the UFO as the hypotenuse, and the height of the UFO as the other leg. Let's denote the horizontal distance from Skippy to the point directly below the UFO as \(x\) feet.

Step 3: Calculate Skippy's Distance to the UFO Base

Using the tangent function for Skippy's angle: \( \tan(75^{\circ}) = \frac{h}{x} \), where \(h\) is the height of the UFO. We solve for \(x\): \( x = \frac{h}{\tan(75^{\circ})} \).

Step 4: Set Up the Right Triangle for Sally's Side

Similarly, for Sally with a 50-degree angle, we can set up: \( \tan(50^{\circ}) = \frac{h}{10560 - x} \).

Step 5: Calculate from Sally's Side

Solving for the unknown distance from Sally's position, we get: \( 10560 - x = \frac{h}{\tan(50^{\circ})} \).

Step 6: Solve the System of Equations

We now have two equations: \( x = \frac{h}{\tan(75^{\circ})} \) and \( 10560 - x = \frac{h}{\tan(50^{\circ})} \). Substituting the expression for \(x\) in the second equation gives: \( 10560 - \frac{h}{\tan(75^{\circ})} = \frac{h}{\tan(50^{\circ})} \).

Step 7: Solve for Height h

Rearranging, we combine and solve for \(h\):\[10560 = \frac{h}{\tan(50^{\circ})} + \frac{h}{\tan(75^{\circ})}\]This simplifies to:\[10560 = h \left( \frac{1}{\tan(50^{\circ})} + \frac{1}{\tan(75^{\circ})} \right)\]Solving for \(h\), we calculate the value, taking care to use the correct trigonometric values:\[h = \frac{10560}{\frac{1}{\tan(50^{\circ})} + \frac{1}{\tan(75^{\circ})}}\]

Step 8: Convert Height to Feet

After solving, the value of \(h\) gives the height in feet directly since the original distance 10560 feet was used.

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry is a branch of mathematics that allows us to solve problems involving triangles in which one angle is exactly 90 degrees. This aspect is crucial in problems like Skippy and Sally’s UFO sighting, where the line of sight and the horizontal distance form two sides of a right triangle. Each component of a right triangle plays a part in determining other unknown values of the triangle using trigonometric functions such as sine, cosine, and tangent.
In such problems, understanding which side represents the adjacent, opposite, and hypotenuse in relation to an angle can help apply the correct trigonometric ratio. For instance, when Skippy sees the UFO, his horizontal distance to the UFO’s base forms the adjacent side of the triangle, while the height of the UFO acts as the opposite side.

Angle of Inclination

The angle of inclination is the angle formed between the horizontal line and the line of sight to the object being viewed, in this case, the UFO. This angle is pivotal because it helps us determine the relationships between the sides of the triangle using trigonometry.
When Skippy records an angle of inclination of 75°, it indicates that the UFO is being observed at a steep angle compared to the ground. Similarly, Sally's 50° angle shows a less steep view. Knowing these angles allows us to leverage trigonometric functions to establish equations that will help find the desired height of the UFO.

• The angle of inclination is always measured with respect to the horizontal ground.
• The larger the angle of inclination, the closer or higher the object seems to be.

Tangent Function

The tangent function is a fundamental concept in trigonometry, which relates an angle in a right triangle to the ratio of the opposite side and the adjacent side. For Skippy and Sally’s UFO observation, the tangent function enabled setting up the equations needed to find the height of the UFO.
Mathematically, it is expressed as:

• For Skippy: \( \tan(75^{\circ}) = \frac{h}{x} \)
• For Sally: \( \tan(50^{\circ}) = \frac{h}{10560 - x} \)

These equations play a pivotal role as they connect the observed angles (inclinations) and the dimensions of each right triangle involved. By manipulating these tangent ratios, we are able to solve for the missing height, which remains unknown.

Trigonometric Equations

Trigonometric equations involve relationships between angles and the sides of a triangle using trigonometric functions such as sine, cosine, and tangent. In the UFO problem, we derive two separate equations based on the angles of inclination and solve them simultaneously.
The process involves:

• Deriving an expression for each person viewing the UFO—Skippy and Sally—in terms of known values and the height \( h \).
• For Skippy: \( x = \frac{h}{\tan(75^{\circ})} \)
• For Sally: \( 10560 - x = \frac{h}{\tan(50^{\circ})} \)

By substituting and equating these to the overall distance, we establish a single equation that allows solving for \( h \). This application of trigonometric equations illustrates how to deal with multiple perspectives on the same distance problem to find a singular, consistent solution.