Problem 72 - Precalculus

Question

Are based on the idea of a glide slope (the angle the flight path makes with the ground). Precision Approach Path Indicator (PAPI) lights are used as a visual approach slope aid for pilots landing aircraft. A typical glide path for commercial jet airliners is \(3^{\circ} .\) The space shuttle has an outer glide approach of \(18^{\circ}-20^{\circ} .\) PAPI lights are typically configured as a row of four lights. All four lights are on, but in different combinations of red or white. If all four lights are white, then the angle of descent is too high; if all four lights are red, then the angle of descent is too low; and if there are two white and two red, then the approach is perfect. (IMAGE CANNOT COPY) If the same pilot in Exercise 71 raises the nose of the gliding shuttle so that she drops only 500 feet by the time she is 7800 feet from the shuttle landing strip (ground distance), what is her glide angle then (round to the nearest degree)? Is she within the specs to land the shuttle?

Step-by-Step Solution

Answer

The calculated glide angle is approximately \(3.66^{\circ}\), which is not within the \(18^{\circ} - 20^{\circ}\) range required for the shuttle's glide approach.

Step 1: Understand the Glide Path Scenario

The pilot is aiming for a correct glide path, indicated by two red and two white PAPI lights. We need to determine if the new descent angle achieved by dropping 500 feet while 7800 feet from the landing strip meets this condition.

Step 2: Use Trigonometry to Find the Glide Angle

The glide angle can be calculated using the tangent function, since tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, the 'opposite' side is the altitude change (500 feet) and the 'adjacent' side is the ground distance (7800 feet).Calculate: \[ \tan(\theta) = \frac{500}{7800} \]Solving for \(\theta\): \( \theta = \arctan\left(\frac{500}{7800}\right) \)

Step 3: Calculate the Angle

Compute \( \theta = \arctan\left(\frac{500}{7800}\right) \) using a calculator: \( \theta \approx \arctan(0.0641) \approx 3.66^{\circ} \).

Step 4: Determine if the Glide Angle is Within Shuttle Specs

Compare the calculated glide angle of \(3.66^{\circ}\) with the outer glide approach specs for the space shuttle, which are \(18^{\circ} - 20^{\circ}\). Since \(3.66^{\circ}\) falls well below this range, the shuttle is not within specs for a correct approach.

Key Concepts

Understanding the Glide Slope

The glide slope refers to the path an aircraft follows when descending to land. It is an angle that represents the descent's steepness related to the ground.
Glide slopes are crucial for ensuring a safe approach to the runway.

• For commercial airliners, a typical glide slope is around 3 degrees.
• The space shuttle, due to its unique flight dynamics, has a much steeper glide slope, varying between 18 and 20 degrees.

Glide path aids, such as the Precision Approach Path Indicator (PAPI), help pilots maintain the correct glide slope.
These visual aids use a combination of red and white lights to indicate whether the aircraft is on the correct descent path. If the lights show:

• All white: the descent angle is too high.
• All red: the descent angle is too low.
• Two red and two white: the descent angle is optimal.

When approaching to land, pilots rely on these indicators to ensure they do not descent too quickly or too slowly.

Using the Tangent Function in Trigonometry

The tangent function is a fundamental concept in trigonometry used to relate angles to side lengths in right triangles.
Specifically, for a given angle in a right triangle, the tangent is the ratio of the length of the opposite side to the length of the adjacent side.
In the context of a glide slope:

• "Opposite side" refers to the altitude change (how much the aircraft descends).
• "Adjacent side" is the horizontal ground distance covered during descent.

To calculate the glide angle, you can use the equation:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]This equation can be rearranged to solve for the angle \(\theta\) using the inverse tangent function:\[\theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)\]Using a calculator, you can find the angle's value, as done in the exercise to check the shuttle's descent specifications.

What is a Right Triangle?

A right triangle is a type of triangle that has one angle measuring 90 degrees. This geometric shape is crucial in trigonometry because it's the basis for definitions and calculations of trigonometric functions like sine, cosine, and tangent.
Key features of a right triangle:

• The side opposite the right angle is called the hypotenuse, and it's the longest side.
• The other two sides are referred to as the adjacent (next to the angle in question) and opposite (across from the angle in question) sides.

In the context of an aircraft's glide slope:

• The altitude change acts as the 'opposite' side.
• The horizontal ground distance traveled acts as the 'adjacent' side.

Understanding right triangles is essential for effectively using trigonometric functions to solve real-world problems involving angles and distances, such as determining an aircraft's glide angle.

Angle of Descent and Its Importance

The angle of descent is the angle at which an aircraft decreases altitude while approaching a runway. It's crucial for maintaining a safe and controlled landing procedure.
Pilots need to be keenly aware of this angle to:

• Ensure they are neither descending too steeply nor too shallowly.
• Maintain the structural integrity and safety of the aircraft.
• Effectively match the landing specifications unique to each type of aircraft.

In the example provided:

• The calculated angle of descent was approximately 3.66 degrees.
• Compared to the space shuttle's typical descent specs of 18 to 20 degrees, it was much lower.

This indicates the shuttle pilot needs to adjust her glide path to match the correct descent angle, ensuring a safe and aligned approach to the landing strip.
Understanding and calculating the angle of descent helps pilots make informed decisions during critical phases of flight.