Related Rates - Angle of Elevation Problem

The Organic Chemistry Tutor

2 Mar 201811:36

EducationalLearning

32 Likes 10 Comments

TLDRThe video script presents a mathematical problem involving an airplane flying horizontally at 500 miles per hour at an altitude of three miles. The objective is to calculate the rate at which the angle of elevation changes for an observer on the ground as the plane approaches. Using trigonometric principles and the tangent function, the video demonstrates how to determine the distance the plane travels horizontally (x) and then applies differentiation to find the rate of change of the angle (d theta/dt). The solution is methodically derived, resulting in a rate of change in radians per hour, with additional conversions provided for per minute and per second for different unit preferences.

Takeaways

🛫 The airplane is traveling horizontally at a speed of 500 miles per hour.
🚁 The plane's altitude above the ground is constant at three miles.
🧭 The observer measures an angle of elevation of 60 degrees.
📐 The problem involves calculating the rate of change of the angle of elevation (dθ/dt).
📈 The value of x (horizontal distance) decreases as the plane approaches the observer, with dx/dt being -500 mph.
📍 The relationship between x, y, and θ is given by the tangent function: tan(θ) = y/x, with y being constant at 3 miles.
🔢 Using the 30-60-90 triangle properties, tan(60°) = √3, helping to determine the value of x.
🤔 The derivative of the tangent function with respect to time is used to find dθ/dt.
📚 The constant multiple rule and power rule of differentiation are applied to the tangent equation.
🌟 The final calculation results in a rate of change of the angle of elevation of 125 radians per hour.
🔄 The rate can be converted to radians per minute or per second if needed, using time unit conversions.

Q & A

What is the speed of the airplane mentioned in the script?
-The airplane is traveling at a speed of 500 miles per hour.
At what altitude is the airplane flying above the ground?
-The airplane is flying at an altitude of three miles above the ground.
What is the angle of elevation at which the observer sees the plane?
-The observer sees the plane at an angle of elevation of 60 degrees.
What is the relationship between the variables x, y, and z in the context of the problem?
-In the context of the problem, x represents the distance along the ground towards the plane, y is the altitude of the plane (which is constant at 3 miles), and z would be the hypotenuse of the right triangle formed with x and y on the ground.
How is the value of x calculated using the given angle of elevation and altitude?
-The value of x is calculated using the tangent of the angle of elevation (theta), which is the ratio of the opposite side (y) to the adjacent side (x). Since tan(60 degrees) equals the square root of 3, and y is 3 miles, x is calculated as 3 divided by the square root of 3, which simplifies to the square root of 3.
What is the rate at which the angle of elevation (d theta/dt) is changing, and what are its units?
-The rate at which the angle of elevation is changing, d theta/dt, is 125 radians per hour. If converted to seconds, it would be approximately 0.0347 radians per second, or in minutes, it would be 2.08 radians per minute.
Why is dx/dt negative in this scenario?
-dx/dt is negative because as the plane travels towards the observer, the distance x along the ground is decreasing.
How is the derivative of the tangent function related to the secant function?
-The derivative of the tangent function with respect to an angle is the secant function squared times the angle. In mathematical terms, d/dt(tan(theta)) = secant^2(theta) * d theta/dt.
What is the significance of the 30-60-90 triangle in solving this problem?
-The 30-60-90 triangle is significant in this problem because it provides the ratios of the sides opposite to the angles in the triangle. In this case, the tangent of 60 degrees is the square root of 3, which is used to find the value of x. Additionally, the cosine of 60 degrees, which is 1/2, is used in the final calculation of d theta/dt.
How is the constant multiple rule applied in differentiating the expression y/x with respect to time?
-The constant multiple rule states that the derivative of a constant multiplied by a function is the constant times the derivative of the function. In this case, y is treated as a constant (since it represents the altitude which does not change), and x is the variable. The derivative of y/x with respect to time is then calculated as y times the derivative of x to the power of -1 (which is -1/x) times dx/dt.
What is the role of the power rule in differentiating the expression x to the power of -2?
-The power rule states that the derivative of x to the power n is n*x to the power (n-1). When differentiating x^(-2) with respect to time, the power rule gives us -2*x^(-3), which is then multiplied by dx/dt to find the rate of change of the angle of elevation.
How can the negative value of dx/dt be interpreted in the context of the airplane's movement?
-The negative value of dx/dt indicates that the distance x is decreasing as time progresses. This is interpreted as the airplane moving closer to the observer on the ground, hence reducing the distance along the ground from the observer to the plane.

Outlines

00:00

🛫 Introduction to the Airplane's Motion and Angle of Elevation

This paragraph introduces the scenario of an airplane flying horizontally at a speed of 500 miles per hour at an altitude of three miles above the ground. An observer on the ground sees the plane approaching at an angle of elevation of 60 degrees. The main objective is to calculate the rate at which the angle of elevation (dθ/dt) is changing at this specific moment. The paragraph sets up the problem by describing the situation visually and identifying the known quantities: the altitude (y = 3 miles), the speed of the plane (500 mph), and the angle of elevation (θ = 60 degrees). It also introduces the variables x (horizontal distance) and dx/dt (the rate at which x is changing), which will be crucial for the calculations to follow.

05:01

📐 Using Trigonometry to Determine the Variables

In this paragraph, the focus is on using trigonometry to determine the values of x and the rate of change of the angle of elevation (dθ/dt). The paragraph explains that since y is constant (as the plane maintains its altitude), the relationship between x, y, and θ can be expressed using the tangent function. By using the tangent of 60 degrees, which is equal to the square root of 3, the paragraph calculates the value of x to be the square root of 3. It then proceeds to differentiate the tangent equation with respect to time to find dθ/dt. The process involves applying the chain rule and the power rule, taking into account that dx/dt is negative 500 miles per hour (as x decreases). The paragraph emphasizes the importance of understanding trigonometric identities and the 30-60-90 triangle to solve the problem.

10:04

🕒 Conversion of Units for Rate of Change

The final paragraph focuses on converting the calculated rate of change of the angle of elevation (dθ/dt) from radians per hour to other units such as radians per minute and radians per second. It begins by explaining the conversion of hours to minutes and then to seconds, emphasizing the cancellation of units to arrive at the desired unit. The paragraph provides the calculations for converting 125 radians per hour to 2.08 radians per minute and 0.0347 radians per second. It concludes by suggesting that the final answer should match the units of the given speed (miles per hour), and thus, the rate of change is best represented in radians per hour.

Mindmap

Keywords

💡Airplane

An airplane is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine or propeller. In the video, the airplane is the central object of study, traveling horizontally at a speed of 500 miles per hour. Its motion and interaction with the ground observer form the basis of the mathematical problem being solved.

💡Altitude

Altitude refers to the height of an object or point in relation to a reference plane, typically the ground or sea level. In the context of the video, the altitude of the airplane is a critical piece of information that is used to determine the angle of elevation and the rate at which this angle changes over time.

💡Observer

An observer in this context is a person or entity that is watching or monitoring the situation, in this case, the flight of the airplane. The observer's perspective is crucial in calculating the angle of elevation and understanding how the airplane's motion affects this angle.

💡Angle of Elevation

The angle of elevation is the angle between a horizontal line at the observer's location and the line of sight to the object being observed. In the video, the angle of elevation is key to determining the rate at which this angle changes as the airplane approaches the observer.

💡Rate of Change

The rate of change, often denoted as 'd theta dt' in mathematical terms, refers to the rate at which a variable changes with respect to another variable, typically time. In the video, the main objective is to calculate the rate at which the angle of elevation changes as the airplane travels towards the observer.

💡Tangent

The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In the video, the tangent function is used to establish the relationship between the altitude (y) and the horizontal distance (x) in relation to the angle of elevation (theta).

💡Secant

Secant is the reciprocal of the cosine function, and it is used to describe the ratio of the hypotenuse to the adjacent side in a right triangle. In the context of the video, the secant function is used in the differentiation process to find the rate of change of the angle of elevation.

💡Differentiation

Differentiation is a mathematical operation that finds the rate at which a function changes with respect to one of its variables. In the video, differentiation is applied to the tangent function to find the rate of change of the angle of elevation with respect to time.

💡Right Triangle

A right triangle is a triangle in which one of the angles is a right angle (90 degrees). In the video, the relationship between the altitude, horizontal distance, and angle of elevation forms a right triangle, which is used to apply trigonometric principles and solve for the unknowns.

💡Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, particularly right triangles. In the video, trigonometry is essential for solving the problem, as it allows the calculation of unknown sides and angles based on given information.

💡Horizontal

In the context of the video, 'horizontal' refers to the plane or direction that is parallel to the ground or the Earth's surface. The airplane is said to be traveling horizontally, which means it maintains a constant altitude while moving forward.

💡Speed

Speed is a measure of how fast an object is moving, and is typically expressed in units of distance over time. In the video, the speed of the airplane is given as 500 miles per hour, which is used in the calculations to determine the rate of change of the angle of elevation.

Highlights

An airplane is traveling horizontally at a speed of 500 miles per hour.

The airplane is at an altitude of three miles above the ground.

An observer sees the plane flying towards him at an angle of elevation of 60 degrees.

The objective is to find the rate at which the angle of elevation is changing at this instant, denoted as d theta/dt.

The value of y (altitude) is 3 miles, which remains constant during the observation.

The rate at which x (horizontal distance from the observer) is changing, dx/dt, is negative 500 miles per hour, as the plane moves towards the observer.

The tangent of the angle theta (tan(theta)) is equal to the opposite side (y) divided by the adjacent side (x), which is 3 divided by x.

The tangent of 60 degrees is the square root of 3, based on the 30-60-90 triangle properties.

The value of x (horizontal distance) is calculated to be the square root of 3 miles.

The rate of change of the angle of elevation, d theta/dt, is determined using the tangent equation and differentiation with respect to time.

The derivative of the tangent function with respect to theta is secant squared, which is used in the differentiation process.

The derivative of y over x (constant multiple rule) is used to express the relationship between x and y in terms of dx/dt.

Secant squared is calculated as 1 divided by cosine squared, which is utilized in the differentiation of the tangent equation.

Cosine of 60 degrees is one-half, derived from the 30-60-90 triangle properties.

The final calculation for d theta/dt (the rate of change of the angle of elevation) results in 125 radians per hour.

The rate of change of the angle can also be expressed in radians per minute or radians per second, by converting the time units accordingly.

Transcripts

Browse More Related Video

How solve a problem with changing angle for related rates

Related Rates - Angle of Elevation Problem

Takeaways

Q & A

What is the speed of the airplane mentioned in the script?

At what altitude is the airplane flying above the ground?

What is the angle of elevation at which the observer sees the plane?

What is the relationship between the variables x, y, and z in the context of the problem?

How is the value of x calculated using the given angle of elevation and altitude?

What is the rate at which the angle of elevation (d theta/dt) is changing, and what are its units?

Why is dx/dt negative in this scenario?

How is the derivative of the tangent function related to the secant function?

What is the significance of the 30-60-90 triangle in solving this problem?

How is the constant multiple rule applied in differentiating the expression y/x with respect to time?

What is the role of the power rule in differentiating the expression x to the power of -2?

How can the negative value of dx/dt be interpreted in the context of the airplane's movement?