Navigation Word Problem

Khan Academy
8 Dec 200708:00
EducationalLearning
32 Likes 10 Comments

TLDRIn this instructional video, the presenter tackles a problem involving a law enforcement boat and a smuggling craft departing from opposite sides of Lake Michigan. They explain the boating convention of measuring course angles clockwise from due north and use this to determine the directions of both vessels. The law enforcement boat heads out from Milwaukee on a 105-degree course, while the smuggling craft leaves Grand Haven on a 195-degree course. The presenter then constructs a right triangle to represent their paths and uses trigonometry to calculate the unknown speed of the smuggling craft, given the law enforcement boat's speed of 23 knots and the time it took for them to collide.

Takeaways
  • 🗺️ The problem involves two locations, Milwaukee, Wisconsin, and Grand Haven, Michigan, situated on opposite sides of Lake Michigan.
  • 🛳️ A law enforcement boat leaves Milwaukee on a course of 105 degrees, and a smuggling craft leaves Grand Haven on a course of 195 degrees.
  • 🧭 The boating convention used in the script defines a course as the number of degrees clockwise from due north, where due north is 0 degrees.
  • 📏 The law enforcement boat's course of 105 degrees is calculated to be 15 degrees south of west, while the smuggling craft's course of 195 degrees is 15 degrees north of east.
  • 🚤 The two vessels are traveling towards each other, with the law enforcement boat averaging 23 knots.
  • ⏱ Both boats leave at the same time and eventually collide, implying they travel for the same duration, denoted as time 't'.
  • 📐 The problem is essentially about calculating the speed of the smuggling craft using the geometry of the situation, which forms a right triangle.
  • 🔢 The distance traveled by the law enforcement boat is represented as 23t (speed times time), and the distance for the smuggling craft is represented as xt.
  • 📈 The tangent of the angle (75 degrees) between the paths of the two boats is used to relate the distances traveled by each boat.
  • 📚 The formula derived from the tangent function is used to solve for the unknown speed 'x' of the smuggling craft: x = 23 / tan(75°).
  • 🔍 The final step is to calculate the tangent of 75 degrees and then divide 23 by this value to find the smuggling craft's speed, which is expected to be a high number.
Q & A

  • What is the geographical relationship between Milwaukee, Wisconsin, and Grand Haven, Michigan as described in the script?

    -Milwaukee, Wisconsin, is directly west of Grand Haven, Michigan, situated on opposite sides of Lake Michigan.

  • How does the script describe the initial drawing to represent the positions of Milwaukee and Grand Haven?

    -The script describes drawing a horizontal line in blue to represent the water of Lake Michigan, with Milwaukee to the west and Grand Haven to the east.

  • What is the course direction of the law enforcement boat as it leaves Milwaukee?

    -The law enforcement boat leaves Milwaukee on a course of 105 degrees, which is 105 degrees clockwise from due north.

  • How is the course direction of the smuggling craft from Grand Haven described in the script?

    -The smuggling craft steers a course of 195 degrees from Grand Haven, which is calculated as 195 degrees clockwise from due north.

  • What is the significance of the angle calculations in the script?

    -The angle calculations are used to determine the direction each boat is traveling and to set up the problem for solving the right triangle formed by their paths.

  • What is the average speed of the law enforcement boat as mentioned in the script?

    -The law enforcement boat averages 23 knots.

  • How does the script use the concept of tangent to relate the distances traveled by the two boats?

    -The script uses the tangent of the angle (75 degrees) to set up a ratio of the opposite side (23t) over the adjacent side (xt), allowing for the calculation of the unknown speed x of the smuggling craft.

  • What is the relationship between the speeds of the two boats as they travel towards each other?

    -The relationship is that both boats leave at the same time and collide after the same amount of time t, implying that the product of their speeds and time (distance) is equal.

  • What mathematical concept is used to solve for the unknown speed of the smuggling craft?

    -The mathematical concept used is the tangent function from trigonometry, specifically the tangent of an angle in a right triangle.

  • How does the script suggest finding the smuggling craft's average speed?

    -The script suggests using the tangent of 75 degrees and dividing 23 (the speed of the law enforcement boat in knots) by this tangent value to find the smuggling craft's speed x.

  • What is the conclusion about the triangle formed by the paths of the two boats?

    -The conclusion is that the triangle formed by the paths of the two boats is a right triangle, with one angle being 90 degrees, which is deduced from the sum of the other two angles being 180 degrees minus 90 degrees.

Outlines
00:00
🚤 Navigational Problem Setup

The script begins with the presenter addressing a navigational problem involving Milwaukee, Wisconsin, and Grand Haven, Michigan, positioned on opposite sides of Lake Michigan. The presenter visually maps out the geographical locations and introduces the concept of a course angle in boating, explaining it as degrees clockwise from due north. The law enforcement boat from Milwaukee and a smuggling craft from Grand Haven are described as setting off on courses of 105 and 195 degrees, respectively. The presenter calculates the angles relative to the cardinal directions and establishes that the paths of the two vessels will form a right triangle.

05:06
🔍 Solving for the Smuggling Boat's Speed

In the second paragraph, the presenter focuses on solving for the unknown average speed of the smuggling boat. Both the law enforcement boat, traveling at a known speed of 23 knots, and the smuggling craft depart simultaneously and collide at some point, indicating they traveled for the same duration, denoted as 't'. The presenter uses the properties of a right triangle and the tangent function from trigonometry to set up an equation relating the distances traveled by both boats. By applying the tangent of the known angle (75 degrees) to the distances, the presenter derives a formula to solve for the unknown speed 'x' of the smuggling boat, which involves dividing 23 by the tangent of 75 degrees. The presenter suggests using a calculator to find the tangent value and perform the final calculation, thus solving for 'x'.

Mindmap
Keywords
💡Milwaukee
Milwaukee is a city in the U.S. state of Wisconsin. In the video's context, it is used as a starting point for a law enforcement boat's journey. The script describes Milwaukee as being directly west of Grand Haven, Michigan, which sets up the geographical scenario for the problem being discussed.
💡Grand Haven
Grand Haven is a city in the U.S. state of Michigan. In the video, it is mentioned as the location directly east of Milwaukee, across Lake Michigan. It serves as the starting point for a smuggling craft, which is a key element in the navigation problem presented in the video.
💡Lake Michigan
Lake Michigan is one of the Great Lakes of North America and is situated between the states of Wisconsin and Michigan. The video uses Lake Michigan as the body of water separating Milwaukee and Grand Haven, which is central to the navigation scenario being discussed.
💡Course
In the context of the video, 'course' refers to the direction in which a boat is traveling, measured in degrees clockwise from due north. The script explains that a course of 105 degrees for the law enforcement boat and 195 degrees for the smuggling craft are crucial to determining their paths and eventual collision.
💡Degrees
Degrees are units used to measure angles, and in this video, they are used to specify the direction of travel for the boats. The script describes how a course is determined by the number of degrees a boat is steering from due north, which is foundational to solving the navigation problem.
💡Law Enforcement Boat
The law enforcement boat is a vessel in the video that is on a mission to intercept a smuggling craft. It leaves from Milwaukee and travels at a course of 105 degrees at a speed of 23 knots, which is essential information for calculating the distance it travels and its eventual collision with the smuggling craft.
💡Smuggling Craft
The smuggling craft is another vessel in the video that is attempting to evade capture. It leaves from Grand Haven and travels at a course of 195 degrees. The video's problem involves determining the speed of this smuggling craft based on the information provided.
💡Knots
Knots are a unit of speed used in navigation, equivalent to one nautical mile per hour. In the video, the law enforcement boat's speed is given as 23 knots, which is used to calculate the distance it travels before the collision with the smuggling craft.
💡Right Triangle
A right triangle is a triangle with one angle measuring 90 degrees. The video describes the paths of the two boats forming a right triangle, which is key to using trigonometric functions to solve for the unknown speed of the smuggling craft.
💡Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In the video, trigonometric concepts, specifically the tangent function, are used to relate the angles and sides of the right triangle formed by the paths of the two boats to find the unknown speed of the smuggling craft.
💡Collision
In the video, the term 'collision' refers to the point at which the paths of the law enforcement boat and the smuggling craft intersect. The problem involves calculating the time and distance each boat travels until this collision occurs, which is the ultimate goal of the navigation scenario presented.
Highlights

Milwaukee, Wisconsin is directly west of Grand Haven, Michigan on opposite sides of Lake Michigan.

Drawing a horizontal line in blue to represent water between Milwaukee and Grand Haven.

Law enforcement boat leaves Milwaukee on a course of 105 degrees.

Boating convention where course is degrees clockwise from due north.

Calculating the direction of the law enforcement boat's course as 105 degrees clockwise from due north.

Smuggling craft steers a course of 195 degrees from Grand Haven.

Determining the smuggling craft's course as 195 degrees clockwise from due north.

The law enforcement boat averages 23 knots.

Both boats leave at the same time and collide, implying the travel time is the same for both.

Using the formula speed times time equals distance to calculate the distances traveled.

Recognizing the problem involves a right triangle and applying trigonometry.

Using tangent function (SOHCAHTOA) to relate the sides of the right triangle to the angles.

Setting up the equation to solve for the smuggling boat's speed (x) using tangent of 75 degrees.

Solving for x gives the formula x = 23 / tan(75 degrees).

Highlighting the importance of using a calculator to find the tangent of 75 degrees for the final calculation.

Concluding the problem-solving process with the formula for the smuggling boat's speed.

Transcripts

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Boat NavigationMath ProblemLake MichiganWisconsinMichiganCourse AnglesSpeed CalculationCollision CourseTangent RatioUnit CircleDistance Formula