Problem 68 - Precalculus

Question

Illustrate a mid-air refueling scenario that military aircraft often enact. Assume the elevation angle that the hose makes with the plane being fueled is \(\boldsymbol{\theta}=36^{\circ}\). (IMAGE CANNOT COPY) If the smallest acceptable altitude difference \(a\) between the two planes is 100 feet, how long should the hose be? Round to the nearest foot.

Step-by-Step Solution

Answer

The hose should be approximately 170 feet long.

Step 1: Understand the problem setup

In mid-air refueling, two planes are involved: a tanker and a receiving aircraft. The question describes the angle of elevation \(\theta\) as the angle formed between the hose and the horizontal line from the receiving aircraft to the tanker. The task is to find the length of the hose (hypotenuse) that can maintain a minimum altitude difference \(a = 100\) feet between the planes.

Step 2: Set up the trigonometric relation

The situation describes a right triangle where the angle \(\theta = 36^{\circ}\) and the opposite side is the altitude difference \(a = 100\) feet. We need to find the hypotenuse length of this triangle, which will be the length of the hose (\(L\)). We use the sine function: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{L} \]

Step 3: Solve for the hose length \(L\)

Rearrange the sine equation to solve for \(L\):\[ L = \frac{a}{\sin(\theta)} \]Substitute the known values \(a = 100\) feet and \(\theta = 36^{\circ}\):\[ L = \frac{100}{\sin(36^{\circ})} \]

Step 4: Calculate the sine of the angle and the hose length

Calculate \(\sin(36^{\circ})\). Use a calculator to find that \(\sin(36^{\circ}) \approx 0.5878\). Substitute the value of the sine into the equation:\[ L = \frac{100}{0.5878} \approx 170.1 \]

Step 5: Round the length of the hose to the nearest foot

Finally, round the calculated hose length to the nearest foot:\[ L \approx 170 \text{ feet} \]

Key Concepts

Angle of Elevation

In this scenario, the angle of elevation is defined as the angle that forms between the horizontal line from the receiving aircraft to the tanker and the hose used during refueling. This angle is a crucial element in trigonometry because it helps us relate the distance between two points using values like the height and length of elements forming the triangle. When you look upwards at an object from a certain level, the angle your line of sight makes with the horizontal line below is the angle of elevation. This is often used in various real-life applications such as determining the height of towers, the trajectory of missiles, or as in this exercise, the altitude difference between two aircraft.
To measure the angle of elevation, always remember it must be measured from the horizontal line to the line of sight, forming a triangle that usually involves a vertical height or distance as one of the sides.

• Angle of elevation is always measured from the horizontal upward to the object or line.
• It is often complementary to the angle of depression which is measured downwards.
• Used extensively in navigation, surveying, and architecture.

Right Triangle

A right triangle, as the name implies, incorporates a 90-degree angle. In trigonometry, right triangles make calculations simpler by subdividing the play of angles and sides into understandable pieces. When you have a right triangle, it sets the stage for using trigonometric functions like sine, cosine, and tangent that relate the angles to the ratios of the side lengths. In the context of the problem, the refueling operation forms a right triangle between the hose, the altitude difference, and the line across the two points of interest.
The right triangle is composed of three sides:

• The hypotenuse, which is the longest side opposite the right angle.
• The opposite side, which is the side opposite the angle you are focusing on (here it is the altitude difference).
• The adjacent side, which is the side next to the angle of focus that is not the hypotenuse.

Understanding these sides is fundamental. It allows for solving several elements of triangles that might otherwise seem hidden, such as using trigonometric relationships like the sine or cosine function.

Sine Function

The sine function is crucial in solving many problems involving right triangles. It relates one angle of the triangle to the ratio of the length of the opposite side over the hypotenuse. In the refueling scenario, we used the sine function to find the length of the hose needed while maintaining the necessary altitude difference. Formally, the sine of an angle in a right triangle is given by:\[ \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \]By using this trigonometric relationship, you can find missing side lengths when at least one side and one non-right angle are known.

• The opposite side of the angle is the one directly opposite to the angle of concern.
• The hypotenuse is always the side opposite the right angle and the longest side of the triangle.
• Commonly, calculators have built-in functions to compute sine for various angles.

In the example given, rearranging the formula allowed us to solve for the hypotenuse, providing a straightforward way to determine how long the hose should be when given the angle of elevation and the altitude difference. Remember, understanding and applying the sine function can simplify many geometrical measurements and calculations in both theoretical and real-world problems.