Problem 32 - Precalculus: Functions and Graphs

Question

An airplane takes off at a \(10^{\circ}\) angle and travels at the rate of 250 ft/sec. Approximately how long does it take the airplane to reach an altitude of \(15,000\) feet?

Step-by-Step Solution

Answer

It takes approximately 346 seconds to reach an altitude of 15,000 feet.

Step 1: Understand the Problem

We know the airplane takes off at a \(10^{\circ}\) angle and travels at 250 ft/sec. We need to find the time it takes to reach an altitude of 15,000 feet. Since we have a right triangle, the altitude corresponds to the opposite side, and the hypotenuse is the distance traveled. We can use trigonometric functions to solve this.

Step 2: Set Up Trigonometric Equation

Let's denote the altitude (opposite side) as 15,000 feet. We can use the sine function, which relates the opposite side to the hypotenuse: \[ \sin(10^{\circ}) = \frac{\text{altitude}}{\text{hypotenuse}} = \frac{15,000}{ ext{distance}} \]

Step 3: Solve for the Distance Traveled

Rearrange the equation to find the distance traveled (hypotenuse): \[ \text{distance} = \frac{15,000}{\sin(10^{\circ})} \] Calculate \(\sin(10^{\circ})\) and substitute it into the equation. \(\sin(10^{\circ}) \approx 0.1736\), so:\[ \text{distance} \approx \frac{15,000}{0.1736} \approx 86,377.3 \text{ feet} \]

Step 4: Calculate the Time Taken

We have the speed of the airplane as 250 ft/sec and the distance as 86,377.3 feet. Now, use the formula: \[ \text{time} = \frac{\text{distance}}{\text{speed}} \]Substituting the values, we find:\[ \text{time} = \frac{86,377.3}{250} \approx 345.5 \text{ seconds} \]

Step 5: Conclude with Approximate Time

Rounding to the nearest second, it takes approximately 346 seconds for the airplane to reach an altitude of 15,000 feet.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding relationships within triangles, especially right-angled ones. They involve the measurement of angles and their application to finding unknown side lengths in triangles. In a right triangle, we use specific functions such as sine, cosine, and tangent which are based on the ratios of sides.

- **Sine Function (sin)**: This relates the opposite side of the angle to the hypotenuse. - **Cosine Function (cos)**: This involves the adjacent side and the hypotenuse. - **Tangent Function (tan)**: This compares the opposite side to the adjacent side.

These functions are extremely useful when you know one angle and one side of the triangle and need to find other sides or angles. In real-world applications, like navigation and flight paths, they allow for precise calculations of distances or elevations given angles.

Sine Function

The sine function is vital when working with problems involving angles and lengths. It specifically relates an angle of a right triangle to the ratio of the opposite side over the hypotenuse.

For example, if we know the angle of take-off for an airplane and the desired altitude (opposite side), we can find the total distance traveled along the hypotenuse using the sine function:

• The formula is given by: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), where \( \theta \) is the angle of elevation.
• Rearranging for the hypotenuse gives us: \( \text{hypotenuse} = \frac{\text{opposite}}{\sin(\theta)} \)

In real-world terms, if an airplane ascends at a \(10^{\circ}\) angle to reach an altitude, we can utilize this function to determine how far the airplane must have traveled. This understanding assists pilots in navigation and ensures airplanes reach the required altitudes effectively.

Distance and Speed Calculations

Calculating distance and speed is vital in many practical scenarios such as engineering and transportation. When analyzing the motion of an object or vehicle, knowing both the speed at which it travels and the distance covered is essential for determining time.

- **Distance** formula: \( \text{distance} = \text{speed} \times \text{time} \)- **Speed** formula: \( \text{speed} = \frac{\text{distance}}{\text{time}} \)- **Time** formula: \( \text{time} = \frac{\text{distance}}{\text{speed}} \)

For instance, in the case of the airplane, given the calculated distance and the airplane's speed of 250 ft/sec, one can find how long it will take to achieve a specific height by rearranging the formula to solve for time. These calculations are crucial in ensuring that flights are accurately timed and maintain safety protocols.