Use the Angle of Elevation to Find the Distance Between Two flying airplanes | Step-by-Step Tutorial

PreMath
31 Oct 202008:41
EducationalLearning
32 Likes 10 Comments

TLDRThis tutorial video explores a trigonometry word problem involving two airplanes flying at different heights. The challenge is to find the vertical distance between them given the angles of elevation from a ground point. Using trigonometric ratios, specifically the tangent function, the presenter sets up two equations based on the given angles (45 and 60 degrees) and solves for the unknown height 'h'. The final answer reveals a vertical distance of 1690.6 meters between the planes, offering viewers a clear application of trigonometry in real-world scenarios.

Takeaways
  • 📚 The video is a pre-math tutorial focusing on the applications of trigonometry with a word problem involving two airplanes.
  • 🛫 The scenario involves an airplane flying at a height of 4,000 meters above the ground, passing vertically above another airplane.
  • 📐 The angles of elevation from a point on the ground to each airplane are 60 degrees and 45 degrees, respectively.
  • 🔍 The objective is to find the vertical distance between the two airplanes.
  • 📏 The problem introduces the concept of using trigonometric ratios, specifically the tangent function, to solve for the unknown distance.
  • 📈 The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle.
  • ✂️ The script breaks down the problem into two right triangles, one smaller (triangle CDA) and one larger (triangle BDA), each with a different angle and side lengths.
  • 📉 For triangle CDA with a 45-degree angle, the tangent is equal to 1, leading to the equation \( x = 4000 - h \).
  • 📊 For triangle BDA with a 60-degree angle, the tangent is the square root of 3, resulting in the equation \( x = \frac{4000}{\sqrt{3}} \).
  • 🔗 By comparing the two equations, the script derives a relationship between the height \( h \) and the distance \( x \).
  • 📝 The final step involves solving for \( h \) using the derived equation, which simplifies to \( h = 4000 - \frac{4000}{1.732} \).
  • 🎯 The calculated vertical distance between the two airplanes is approximately 1690.6 meters.
Q & A

  • What is the main topic of the video tutorial?

    -The main topic of the video tutorial is the application of trigonometry in solving a word problem involving the vertical distance between two airplanes flying at different heights.

  • What is the height of the top airplane from the ground?

    -The top airplane is flying at a height of 4,000 meters from the ground.

  • What are the angles of elevation from the ground to the two airplanes?

    -The angles of elevation from the ground to the two airplanes are 60 degrees and 45 degrees, respectively.

  • What trigonometric ratio is used to solve the problem involving the smaller triangle (CDA)?

    -The tangent ratio, which is the ratio of the opposite side to the adjacent side, is used to solve the problem involving the smaller triangle (CDA).

  • What is the value of the tangent of a 45-degree angle?

    -The value of the tangent of a 45-degree angle is 1.

  • How is the equation for the smaller triangle (CDA) set up using the tangent ratio?

    -The equation is set up as 1 (tangent of 45 degrees) equals (4,000 - h) divided by x, where x is the adjacent side and h is the unknown vertical distance between the two airplanes.

  • Which trigonometric ratio is used for the larger triangle (BDA)?

    -The tangent ratio is also used for the larger triangle (BDA), with the angle being 60 degrees.

  • What is the value of the tangent of a 60-degree angle?

    -The value of the tangent of a 60-degree angle is the square root of 3.

  • How is the equation for the larger triangle (BDA) derived?

    -The equation is derived by setting the tangent of 60 degrees equal to the opposite side (4,000 meters) divided by the adjacent side (x), resulting in the equation √3 * x = 4,000.

  • What is the final step to find the vertical distance (h) between the two airplanes?

    -The final step is to compare the two equations derived from the tangent ratios, isolate h, and solve for its value.

  • What is the calculated vertical distance (h) between the two airplanes?

    -The calculated vertical distance (h) between the two airplanes is 1690.6 meters.

  • What is the significance of using trigonometry in this word problem?

    -Trigonometry is significant in this word problem as it provides a mathematical method to determine the unknown vertical distance between two points when the angles and one side of the triangles are known.

Outlines
00:00
📚 Introduction to Trigonometry Application in Airplane Height Problem

This paragraph introduces a trigonometry word problem involving two airplanes flying at different heights. The problem requires finding the vertical distance between the two planes when one is at a constant height of 4,000 meters and the angles of elevation from a ground point to each plane are 60 degrees and 45 degrees respectively. A visual representation is provided to help understand the setup, and the focus is on using the tangent trigonometric ratio to solve for the unknown vertical distance 'h'.

05:03
🔍 Detailed Solution Process for Airplane Height Problem

The second paragraph delves into the solution process for the trigonometry word problem. It begins by setting up two equations based on the tangent ratios for the angles given (45 and 60 degrees) and the known height difference (4,000 meters). The equations are derived by considering the smaller right triangle formed by the ground point and the lower airplane, and the larger triangle involving the upper airplane. The solution involves isolating the variable 'x', which represents the horizontal distance from the ground point to the lower airplane, and then using it to find 'h', the vertical distance between the two airplanes. The final step involves comparing the two equations, solving for 'h', and providing the numerical answer, which is approximately 1,690.6 meters. The paragraph concludes with the final answer and an invitation to subscribe for more educational content.

Mindmap
Keywords
💡Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. In the context of the video, trigonometry is used to solve a word problem involving the calculation of the vertical distance between two airplanes based on the angles of elevation from a ground point. The video demonstrates the practical applications of trigonometric ratios in real-world scenarios.
💡Word Problem
A word problem in mathematics presents a situation described in words that requires the application of mathematical concepts to find a solution. In the video, the word problem involves determining the vertical distance between two airplanes flying at different heights, which is solved by setting up and solving trigonometric equations based on the given angles of elevation.
💡Elevation
In the context of the video, 'elevation' refers to the angle of view from a ground-level observer looking upwards at an object in the sky. The problem specifies two angles of elevation, 60 degrees and 45 degrees, which are used to calculate the relative positions of the two airplanes.
💡Tangent
The tangent function in trigonometry is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In the video, the tangent function is used to relate the angles of elevation to the known and unknown distances in the triangles formed by the airplanes and the observer on the ground.
💡Right Triangle
A right triangle is a triangle that has one angle measuring 90 degrees. The video script discusses two right triangles formed by the lines of sight from the ground to the two airplanes. These triangles are essential for applying trigonometric ratios to find the vertical distance between the airplanes.
💡Adjacent Side
In a right triangle, the adjacent side is the side next to the angle being considered. In the video, the adjacent side is used in the tangent ratio to represent the horizontal distance from the observer to the point directly below the airplane.
💡Opposite Side
The opposite side in a right triangle is the side that is opposite to the angle being considered. In the script, the opposite side represents the vertical distance from the observer to the airplane, which is part of the calculation to find the vertical distance between the two airplanes.
💡Angle of 45 Degrees
An angle of 45 degrees is a specific angle measurement used in the problem to describe the angle of elevation from the ground to one of the airplanes. The script uses this angle to set up an equation involving the tangent function, which simplifies to a 1:1 ratio due to the properties of a 45-degree angle in a right triangle.
💡Angle of 60 Degrees
Similar to the 45-degree angle, the 60-degree angle is another angle of elevation used in the problem. The tangent of a 60-degree angle is the square root of 3, which is used in the script to set up another equation to solve for the unknown distance between the two airplanes.
💡Cross Multiply
Cross multiplying is a method used in algebra to solve equations involving fractions. In the video, cross multiplying is used to clear the fractions in the tangent equations, allowing for the straightforward comparison and solution of the equations to find the vertical distance between the airplanes.
💡Final Answer
The final answer in the context of the video is the calculated vertical distance between the two airplanes, which is the solution to the word problem presented. The script concludes with the calculation of this distance, providing a clear conclusion to the problem-solving process.
Highlights

Introduction to the application of trigonometry in a word problem involving two airplanes and their relative positions.

Description of the problem setup with one airplane flying at 4,000 meters above the ground and passing vertically above another.

Identification of the angles of elevation from a point on the ground to the two airplanes as 60 degrees and 45 degrees.

Introduction of the goal to find the vertical distance between the two airplanes.

Visual representation of the problem with a vertical depiction of the two airplanes and the ground.

Explanation of the distance between the two planes denoted as 'h'.

Introduction of the smaller right triangle CDA and its sides' relationship to the trigonometric ratios.

Use of the tangent trigonometric ratio to relate the angle of 45 degrees to the sides of triangle CDA.

Calculation of the tangent of 45 degrees as 1, leading to the equation 1 = (4000 - h) / x.

Derivation of the first equation (Equation 1) to find the value of 'x'.

Shifting focus to the larger triangle B DA and its tangent ratio.

Application of the tangent of 60 degrees to the sides of triangle B DA.

Derivation of the second equation (Equation 2) using the tangent of 60 degrees.

Comparison of the two derived equations to solve for 'h'.

Isolation of 'h' and simplification of the equation to find the vertical distance.

Final calculation of 'h' using the approximation of the square root of 3 as 1.732.

Conclusion with the final answer of 1690.6 meters as the vertical distance between the two airplanes.

Encouragement for viewers to subscribe for more educational content.

Transcripts

Browse More Related Video

Bearing and Distance | Lesson 1

Trigonometry - How To Solve Right Triangles

Trigonometry Basics

Ferris Wheel Trigonometry Problem

How to Use the Law of Sines in Trigonometry (Precalculus - Trigonometry 32)

Angle of Elevation and Depression Word Problems Trigonometry, Finding Sides, Angles, Right Triangles

Rate This

5.0 / 5 (0 votes)

Thanks for rating:

Related Tags
TrigonometryWord ProblemAirplane HeightAngles of ElevationTangent RatioRight TriangleTrigonometric RatiosVertical DistanceMath TutorialEducational Video