Problem 10 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. Sketch an appropriate figure, unless the figure is given. On a test flight, during the landing of the space shuttle, the ship was \(325 \mathrm{ft}\) above the end of the landing strip. If it then came in at a constant angle of \(6.5^{\circ}\) with the landing strip, how far from the end of the landing strip did it first touch ground? (A successful reentry required that the angle of reentry be between \(5.1^{\circ}\) and \(7.1^{\circ} .\) )

Step-by-Step Solution

Answer

The shuttle first touches ground approximately 2858 feet from the end of the landing strip.

Step 1: Understanding the problem and sketching the figure

We have a right triangle representing the landing path of the shuttle. The height of the shuttle above the landing strip is the opposite side (325 ft), the distance along the landing strip to where the shuttle touches ground is the adjacent side, and the angle of descent is the angle with the ground (\(6.5^{\circ}\)). You should sketch a right triangle with these components labeled.

Step 2: Setting up the trigonometric relationship

To find the distance on the ground (adjacent side), we'll use the tangent function which relates the opposite side (height) and adjacent side (ground distance) in a right triangle. The tangent of the angle \(\theta\) is defined as the opposite side divided by the adjacent side:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]Substitute \(\theta = 6.5^{\circ}\) and opposite = 325 ft into the equation:\[\tan(6.5^{\circ}) = \frac{325}{\text{distance}}\].

Step 3: Solving for the ground distance

Rearrange the equation to solve for the adjacent side (distance):\[\text{distance} = \frac{325}{\tan(6.5^{\circ})}\]Calculate this value using a calculator. Make sure the calculator is set to degree mode when inputting the angle.

Step 4: Confirming calculation and interpreting terms

Perform the calculation to find the distance:\[\text{distance} = \frac{325}{\tan(6.5^{\circ})} \approx 2857.98 \text{ ft}\]Ensure the calculated angle and distance fulfills the requirement for successful reentry, given the angle constraints of 5.1° to 7.1°.

Key Concepts

Right Triangle

The right triangle is a fundamental shape in trigonometry, characterized by having one angle that is precisely 90 degrees. In the context of the space shuttle landing problem, the right triangle helps visualize the relationship between the shuttle's altitude (opposite side), the distance along the runway (adjacent side), and the angle of descent.

• The opposite side represents the vertical line from the shuttle down to the runway, which is given as 325 feet.
• The adjacent side is the horizontal distance from the point directly below the shuttle to the actual point of touchdown on the runway.
• The angle of descent is the acute angle between the runway and the line connecting the shuttle to the runway.

Sketching a right triangle ensures clarity, as it visually breaks the problem down, making it easier to apply trigonometric functions such as tangent.

Tangent Function

The tangent function (\( an\)) is one of the key trigonometric functions you will encounter. It is especially useful in problems involving right triangles where you have an angle and one side length, allowing you to find unknown sides.
The formula for the tangent function relevant to this problem is:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]Here:

• \(\theta\) is the angle of descent, which is 6.5 degrees.
• The opposite side is the height of 325 feet.
• The adjacent side is the distance along the landing streak, which we need to find.

Using the tangent function effectively transforms a potentially complex problem into a simple arithmetic calculation, once the proper trigonometric relationship is established.

Angle of Descent

The angle of descent provides insight into the steepness of the landing path. It is crucial because it dictates how quickly descends over the given displacement along the runway. For successful landings, keeping this angle within certain limits is essential because it affects both the shuttle's speed and path.
In this problem, the provided angle is 6.5 degrees, comfortably fitting the required range between 5.1 and 7.1 degrees. This means the angle was appropriate for a safe landing, falling between the predefined thresholds.
Understanding the angle of descent helps predict how far the ground should be from the initial descent point and ensures an effective and safe landing approach. Making adjustments to this angle in real-world scenarios could potentially enable more successful landings.

Trigonometric Relationships

Trigonometric relationships form the backbone of calculating unknown distances and angles in right triangles. Knowing these relationships unlocks a series of solutions to solve right triangle problems, similar to the shuttle landing question. Besides the tangent function, two other primary trigonometric ratios are:

• Sine (\( ext{sin}\)): \(\text{sin}(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine (\( ext{cos}\)): \(\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)

These functions work together to provide a comprehensive understanding of the internal proportions of right triangles.
In practical terms, knowing which trigonometric relationship to apply for a given problem is key. For scenarios in which only the opposite and adjacent sides are known, the tangent function is your best ally, as demonstrated in this shuttle landing exercise. Understanding these trigonometric relationships can thus be hugely beneficial not only for homework problems but also for a wide range of real-world applications.