Problem 37 - Thinking Mathematically

Question

A plane rises from take-off and flies at an angle of \(10^{\circ}\) with the horizontal runway. When it has gained 500 feet in altitude, find the distance, to the nearest foot, the plane has flown.

Step-by-Step Solution

Answer

To find the distance flown, use the formula \(d =\frac{500}{\sin(10^{\circ})}\). It is approximately 2897 feet.

Step 1: Define Known Variables

The plane's angle of ascent is \(10^{\circ}\) and its altitude is 500 feet. We need to find the distance flown, which we'll denote as 'd'.

Step 2: Use the Sine Function

In this scenario, 'd' is the hypotenuse of the right triangle formed by the plane's flight path, the altitude gain, and the horizontal distance flown. So, we can use the sine function, which for a right triangle is defined as the ratio of the length of the opposite side to that of the hypotenuse. Here, \(\sin(10^{\circ}) = \frac{500}{d}\).

Step 3: Solve For 'd'

Rearranging the equation yields \(d = \frac{500}{\sin(10^{\circ})}\). Computing this will give the distance the plane has flown, to the nearest foot, expressed in feet.

Key Concepts

Understanding the Sine Function

The sine function is one of the primary functions in trigonometry. It helps us relate the angles of a triangle to the lengths of its sides. Specifically, in a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.

• Opposite Side: The side across from the angle in question.
• Hypotenuse: The longest side in a right triangle, opposite the right angle.

In our exercise, we use the sine function to relate the plane's angle of ascent with its altitude (opposite side) and flight distance (hypotenuse). This is expressed in the formula: \[\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]Where \(\theta\) is the angle of ascent (10°), and the opposite side length is 500 feet. This equation allows us to find the hypotenuse, which is the actual path length of the plane.

Right Triangle Basics

A right triangle is a triangle that has one of its angles equal to 90 degrees. The other two angles must also add up to 90 degrees, making right triangles unique in their geometric properties. This type of triangle is crucial in many real-world applications because of its predictable structure.

• Right Angle: An angle of 90 degrees.
• Adjacent Side: The side next to the angle of interest, excluding the hypotenuse.
• Hypotenuse: Always opposite the right angle and the longest side.

In our scenario, the plane's altitude gain forms the opposite side of the triangle, the runway would form the adjacent side, and the path flown by the plane is the hypotenuse.

Exploring Angle of Ascent

The angle of ascent refers to the angle at which an object, such as an aircraft, ascends relative to a horizontal plane like the runway. It's a key factor in determining the distance traveled as the plane climbs to a certain altitude.
When applying trigonometry, knowing the angle of ascent allows us to calculate the other sides of the associated right triangle. A small angle of ascent, such as 10°, often means the plane covers a greater horizontal distance to gain altitude.
For this reason, using trigonometric functions becomes vital. In the exercise, this angle ensures that we relate the height gained directly to the flight path using the sine function.

Distance Calculation in Flight

Calculating the actual distance a plane has flown involves determining the hypotenuse of the right triangle formed by the plane's path. The sine function enables this calculation by expressing altitude gain and the angle of ascent.
To calculate the actual flight path (hypotenuse) when given the altitude (opposite side) and the angle: \[d = \frac{\text{Altitude}}{\sin(\text{Angle of Ascent})}\]Plugging in our values: \[d = \frac{500}{\sin(10^{\circ})}\] Solving this equation gives the precise distance the plane has flown. It's essential in navigation to understand how far a plane has traveled, ensuring safe and efficient flight operations.