Problem 70 - 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 a commercial jetliner is at an altitude of 450 feet when it is 5200 feet from the runway (approximately 1 mile ground distance), what is the glide slope angle? Will the pilot see white lights, red lights, or both?

Step-by-Step Solution

Answer

The glide slope angle is approximately \(4.95^{\circ}\). All lights appear white.

Step 1: Calculate the Glide Slope Angle

To find the glide slope angle, we need to use the right triangle trigonometry. The altitude of the plane is 450 feet (opposite side of the angle) and the horizontal distance to the runway is 5200 feet (adjacent side of the angle). We can use the tangent function, where \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), to find the angle \( \theta \). So, \( \theta = \tan^{-1}\left(\frac{450}{5200}\right) \).

Step 2: Compute the Angle Using Inverse Tangent

Use a calculator to find \( \theta \): \( \theta = \tan^{-1}\left(\frac{450}{5200}\right) \approx \tan^{-1}(0.08654) \approx 4.95^{\circ} \).

Step 3: Compare Glide Slope Angle with PAPI Indications

The computed glide slope angle is approximately \(4.95^{\circ}\). Compare this angle with the standard \(3^{\circ}\) glide slope for commercial jetliners. Since \(4.95^{\circ}\) is higher than \(3^{\circ}\), this suggests the angle of descent is too steep, indicating all four PAPI lights would appear white to the pilot.

Key Concepts

right triangle trigonometry

Right triangle trigonometry is a mathematical method to solve problems involving right-angled triangles using the relationships between angles and sides. In the context of flight, it's instrumental in calculating the glide slope angle.
To define it simply, a right triangle has a 90-degree angle, and the side opposite this angle is known as the hypotenuse. When you're focusing on an angle that's not the right angle, the side across from this angle is called the opposite side, and the adjacent side is the side that's next to the angle you're considering but not the hypotenuse.

• The sine function relates the opposite side to the hypotenuse.
• The cosine function relates the adjacent side to the hypotenuse.
• The tangent function relates the opposite side to the adjacent side.

In our exercise, we're using the tangent function because we know the opposite side (the altitude of the plane) and adjacent side (the horizontal distance to the runway). The formula is simple: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}. \] This allows us to calculate the angle \(\theta\).

inverse tangent

The inverse tangent, also called arctan or \(\tan^{-1}\), is a function that helps us determine an angle when the tangent value is known. It's used extensively in trigonometry, especially when you know the lengths of the opposite and adjacent sides, as in our glide slope problem.
The inverse tangent function can be denoted as:
\[\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right).\]
By applying the inverse tangent, you reverse the tangent function. This transformation shifts from determining the side ratios when you know an angle, to finding out the angle itself given a ratio. If you're using a calculator, look for \(\tan^{-1}\) or "arctan" to compute this.
In the exercise, when you calculated:
\[\theta = \tan^{-1}\left(\frac{450}{5200}\right),\]
you found the glide slope angle to be around \(4.95^{\circ}\).

Precision Approach Path Indicator (PAPI)

The Precision Approach Path Indicator or PAPI is a visual aid for pilots during landing. It's an essential aviation tool for indicating if an aircraft is on the correct glide slope. PAPI is usually placed beside runways and consists of four lights that can change between red and white.
The lights signal the following:

• All four lights are white: descent angle is too high.
• All four lights are red: descent angle is too low.
• Two lights red, two lights white: descent is on the ideal glide slope.

In terms of safety, these lights are crucial, especially under poor visibility conditions, ensuring that aircraft approach at the standard angle. For most commercial jets, this angle is set at \(3^{\circ}\). However, if a plane is approaching at a steeper angle, like the \(4.95^{\circ}\) found in our problem, all indicators would be white, signaling to the pilot that adjustments are necessary.