Problem 31 - Precalculus: Functions and Graphs

Question

A rocket is fired at sea level and climbs at a constant angle of \(75^{\circ}\) through a distance of \(10,000\) feet. Approximate its altitude to the nearest foot.

Step-by-Step Solution

Answer

The rocket's altitude is 9,659 feet.

Step 1: Understanding the Problem

A rocket climbs at a constant angle of 75 degrees from sea level and covers a horizontal distance of 10,000 feet. We need to find the altitude, which is the vertical component of the given movement.

Step 2: Applying Trigonometry Principles

To find the altitude, we can use the sine function from trigonometry. The altitude ( y ) can be connected to the hypotenuse ( h ) using the sine of the angle, where h = 10,000 feet: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{h} \).

Step 3: Calculating the Altitude

We know: - \( \theta = 75^{\circ} \)- \( h = 10,000 \) feetPlug these values into the formula:\[y = h \cdot \sin(75^{\circ}) = 10,000 \cdot \sin(75^{\circ})\]Using a calculator, \( \sin(75^{\circ}) \approx 0.9659 \). Therefore, \( y = 10,000 \cdot 0.9659 = 9,659 \) feet.

Step 4: Rounding to the Nearest Foot

The calculated altitude y is 9,659 feet, which is already a whole number. Hence, it is rounded to 9,659 feet.

Key Concepts

Understanding the Sine Function

The sine function is one of the basic functions in trigonometry. It helps relate the angles in a right triangle to the lengths of the sides of the triangle. When using the sine function, you deal with an angle and the opposite side and hypotenuse of a right triangle. The sine of an angle \( \theta \) is given by the ratio:

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)

This formula indicates that to find the sine of a specific angle, you take the length of the side opposite to the angle and divide it by the length of the hypotenuse.
For instance, in the rocket problem, the sine function allows us to find the altitude of the rocket—the vertical distance it reaches—by using the angle of the climb and the total distance traveled. By applying \[ y = h \cdot \sin(\theta) \]we can directly calculate the vertical height. The sine function's powerful ability to link angles and side lengths is crucial in solving many real-world problems like this one.

Right Triangle Basics

Right triangles are central to trigonometry because they are simple yet versatile in many problems related to geometry and measurements. A right triangle is characterized by one of its angles being precisely \( 90^{\circ} \). This creates a unique scenario where trigonometry functions like sine, cosine, and tangent can be effectively used.

• Hypotenuse: The longest side opposite the right angle.
• Opposite side: The side opposite to the angle being considered.
• Adjacent side: The side next to the angle being considered.

In the context of the given problem, the rocket's climbing path forms a right triangle with the sea level as the base. The climbing angle given is \( 75^{\circ} \), making it easy to use the sine function to relate the altitude to the hypotenuse. When you understand the roles of the sides and angles, manipulating them with trigonometric functions becomes straightforward.

Angles of Elevation and Their Significance

Angles of elevation are often used when observing something from a lower point, focusing upward toward a height. This angle represents how much you would "tilt" your line of sight upwards from the horizontal base level to view the object.
In any problem involving angles of elevation, it creates a right triangle with:

• The horizontal line as one side (the base).
• The line of sight as the hypotenuse.
• The height or altitude as the opposite side.

For the rocket problem, the angle of elevation is \( 75^{\circ} \). The problem asks for the altitude, which aligns perfectly with this concept. By knowing how to handle angles of elevation, you can solve a wide range of practical problems, whether it's for rockets, ladders leaning against a wall, or even finding the shadow length of an object based on the sunlight angle. Grasping this concept not only allows you to tackle more complex trigonometrical problems but also prepares you to visualize and solve real-world scenarios more effectively.