Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

When an airplane leaves the runway, its angle of climb is \(18^{\circ}\) and its speed is 275 feet per second. Find the plane's altitude after 1 minute.

Step-by-Step Solution

Answer

The plane's altitude after 1 minute is approximately 5156 feet.

Step 1: Understand trigonometric identity

Since the plane moves in a direction of an \(18^{\circ}\) angle, we can use this angle and the speed of the plane to create a right triangle. Here, speed creates the hypotenuse, altitude to be found is the opposite side to the angle, and the sine of the angle is the ratio of the opposite side (altitude) to the hypotenuse (speed). So, we identify that the sine of 18 degrees equals to the altitude divided by the speed.

Step 2: Substitution

Substitute given figures into the equation. We know that the speed is 275 feet per second. But our time unit is different—1 minute—so let's convert it to seconds. There are 60 seconds in 1 minute, so the plane will have flown for 275 feet/second * 60 seconds = 16500 feet. Now, we have: sin(18) = altitude / 16500 feet.

Step 3: Solve for altitude

To find the altitude, we isolate the term 'altitude' on one side of the equation. Altitude = sin(18) * 16500 feet. By calculating this expression, we find the altitude.

Key Concepts

Angle of Climb

The angle of climb is a crucial concept in aviation. It refers to the angle at which an airplane ascends relative to the horizontal. This angle is a measure of how steeply or gently an airplane climbs. In our exercise, the airplane has an angle of climb of \(18^{\circ}\). A smaller angle indicates a gentler ascent, while a larger angle means a steeper climb.

Understanding the angle of climb is important for pilots as it affects the plane’s speed, fuel consumption, and safety. For instance, a steep angle might lead to inefficiencies and instability, whereas a gentle climb keeps the aircraft more efficiently aligned with aerodynamic forces.

In mathematical terms, angles are essential for forming right triangles, where the angle of climb helps in determining other unknowns in the triangle using trigonometric functions.

Sine function

The sine function (\(\sin(\theta)\)) is a fundamental trigonometric function. It is a ratio that helps relate the angles and lengths of sides in a right triangle. Specifically, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

In our example, we used the sine function to find the altitude of the airplane. With a sine value and knowing the hypotenuse (the distance traveled by the plane in 1 minute), we can calculate the length of the opposite side, which is the altitude.

• Formula: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Example used: \( \sin(18^{\circ}) = \frac{\text{altitude}}{16500 \text{ feet}} \)

Using the right value of sine for \(18^{\circ}\), we multiply by 16500 to find the altitude. Understanding how to use the sine function is essential for solving various real-world problems involving triangles and angles.

Right Triangle

A right triangle is a triangle that has one angle measuring \(90^{\circ}\), which is known as a right angle. It forms the basis for many trigonometric calculations. In the context of our exercise, the right triangle is formed by the airplane's path, the altitude (vertical leg), and the horizontal distance.

Key features of a right triangle include:

• One of the angles is always \(90^{\circ}\).
• It consists of two legs and a hypotenuse, where the hypotenuse is the longest side.
• Trigonometric functions like sine, cosine, and tangent are defined specifically for right triangles.

Applying these features, we used the sine function to determine the length of the vertical leg (altitude) based on the known hypotenuse (the flown distance). Right triangles are essential in navigation, physics, and geometry for solving problems involving angles and distances.