Problem 28 - Precalculus with Calculus Previews

Question

Plane with an Altitude Angles of elevation to an airplane are measured from the top and the base of a building that is \(20 \mathrm{~m}\) tall. The angle from the top of the building is \(38^{\circ},\) and the angle from the base of the building is \(40^{\circ} .\) Find the altitude of the airplane.

Step-by-Step Solution

Answer

The altitude of the airplane is approximately 75.2 meters.

Step 1: Understand the Problem

We are given a building of height 20 m. From the top of this building, the angle of elevation to an airplane is 38 degrees, while from the base, it is 40 degrees. We need to find the altitude of the airplane above the ground.

Step 2: Define the Triangles

There are two right-angled triangles formed by the line of sight to the airplane: one from the base and one from the top of the building. The difference in the height of these triangles gives the altitude of the airplane above the top of the building.

Step 3: Use Trigonometry for Triangle 1

For the triangle from the base of the building, we use the tangent function: \[\tan(40^{\circ}) = \frac{h_1}{x}\]where \(h_1\) is the altitude of the airplane above the ground and \(x\) is the horizontal distance to the airplane.

Step 4: Use Trigonometry for Triangle 2

For the triangle from the top of the building, the tangent function gives us:\[\tan(38^{\circ}) = \frac{h_2}{x}\]where \(h_2\) is the height of the airplane above the top of the building. Thus, the total height \(h_1 = h_2 + 20\).

Step 5: Solve the System of Equations

We have two equations:\[ h_1 = x \tan(40^{\circ}) \]\[ h_2 = x \tan(38^{\circ}) \]Substituting \(h_1 = h_2 + 20\) into the first equation gives us:\[ h_2 + 20 = x \tan(40^{\circ}) \]\[ h_2 = x \tan(38^{\circ}) \]Subtract the second equation from the first to find:\[ x ( \tan(40^{\circ}) - \tan(38^{\circ}) ) = 20 \]

Step 6: Calculate Horizontal Distance x

Rearrange to find \( x \):\[ x = \frac{20}{ \tan(40^{\circ}) - \tan(38^{\circ}) } \]Calculate this using a calculator to find the value of \( x \).

Step 7: Calculate Altitude h_1

Substitute the value of \( x \) back into either equation for \( h_1 \):\[ h_1 = x \tan(40^{\circ}) \]This will give the altitude of the airplane above the ground.

Key Concepts

Angle of Elevation

Let's explore the concept of the Angle of Elevation with a simple example. Imagine standing on the ground, looking up at a bird in a tree. The angle your gaze travels upward from the horizontal ground line to the bird is known as the angle of elevation. It helps us determine distances or heights when objects are viewed at an upward angle from a specific point. This angle is crucial in many real-world applications, such as in architecture, aviation, and navigation. In our problem, the angles of elevation taken from different heights help determine the airplane's altitude. By understanding these angles, they assist in drawing imaginary right-angled triangles, essential for calculating distances and heights.

Right-Angle Triangle

A right-angled triangle is a fundamental concept in trigonometry. It contains a 90-degree angle, which makes it a powerful tool for solving problems involving angles and distances. The triangle includes three sides: the hypotenuse, which is the longest side, and two legs, one of which forms the right angle.

• The side opposite the angle of elevation is often the opposite side.
• The side adjacent to the angle is the adjacent side.

In our scenario, two right-angled triangles form from the points of view from both the top and base of the building towards the airplane. These triangles allow us to use trigonometric functions to solve for unknown sides such as the horizontal distance and the altitude of the airplane.

Tangent Function

The tangent function is one of the primary trigonometric functions useful in right-angled triangles. When you know one angle and need to find the length of a side in a right-angled triangle, the tangent function is very helpful.The tangent (\( an(θ)\)) of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side:\[\tan(θ) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]In our airplane example, each triangle can be approached by the tangent function due to the known angles (38° and 40°) and the unknown distances that need solving. By setting up these equations, the tangent of the angles allows us to relate the distances (opposite and adjacent) in the triangles to find the airplane's altitude from the building's top.

Altitude Calculation

Calculating the altitude of an airplane involves finding the vertical distance from the ground to the airplane. In our exercise, we use trigonometric functions and right-angle triangles to solve for altitude mathematically. Starting with forming two equations from the given angle of elevation and known building height:

• The first equation is formed from the base of the building.
• The second equation is from the building's top.

We solve an equation system with both triangles. By finding the horizontal distance first, we can substitute back to find the overall height from the ground to the airplane. This step-by-step use of trigonometry and algebra offers a methodical approach to finding the needed altitude, emphasizing the use of known angles and tangent relationships.