Related Rates - The Baseball Diamond Problem

The Organic Chemistry Tutor
2 Mar 201806:51
EducationalLearning
32 Likes 10 Comments

TLDRThis video script tackles the baseball diamond problem, involving a player running from second to third base. It calculates the rate of change of the player's distance to home plate using the Pythagorean theorem and derivatives. With the player's speed at 20 ft/s and the baseball diamond's square shape with 120 ft sides, the problem forms a right triangle. The solution finds the player's rate of approach to home plate is approximately -7.69 ft/s, indicating the distance is decreasing at this rate as the player moves towards third base.

Takeaways
  • 🏟️ The problem revolves around calculating the rate of change of distance between a baseball player and home plate while moving from second to third base.
  • 🏃‍♂️ The player is running at a constant speed of 20 feet per second.
  • 📏 The baseball diamond forms a square with side lengths of 120 feet.
  • 📍 The player's initial distance from the third base is 50 feet.
  • 🔢 The goal is to find the derivative, dz/dt, which represents the rate of change of the distance, z, between the player and home plate with respect to time.
  • 📐 A right triangle is formed by the player's position relative to the bases, which is used to apply the Pythagorean theorem.
  • 🧮 The calculation starts with determining the value of z (distance from the player to home plate) using the equation z^2 = x^2 + y^2, where x is 50 and y is 120.
  • 🌟 The value of z is found to be 130 feet by taking the square root of the sum of x^2 and y^2.
  • 🎓 The derivative of the equation z^2 = x^2 + y^2 with respect to time is used to find dz/dt, considering x and z as variables and y as a constant.
  • 🔄 The derivative calculation involves the values dx/dt (-20 feet per second) and dy/dt (0, since y does not change during the player's movement from second to third base).
  • 📈 The final result of dz/dt is -100/13 feet per second, or approximately -7.69 feet per second, indicating the rate at which the player is approaching the home plate.
Q & A

  • What is the problem being discussed in the video?

    -The problem discussed in the video is the baseball diamond problem, which involves calculating the rate at which the distance between a player and home plate is changing as the player runs from second base to third base.

  • What is the speed of the baseball player?

    -The baseball player is running at a speed of 20 feet per second.

  • How far is the player from third base at the beginning of the problem?

    -The player is 50 feet away from third base at the beginning of the problem.

  • What is the shape and dimensions of the baseball diamond field?

    -The baseball diamond field has the shape of a square with all side lengths being 120 feet.

  • What is the distance between home plate and third base?

    -The distance between home plate and third base is 120 feet.

  • What is the value of x and y in the right triangle formed by the player's position?

    -In the right triangle formed, x is 50 feet (the distance the player is from third base) and y is 120 feet (the side length of the square diamond field).

  • How is the value of dx/dt determined in this scenario?

    -The value of dx/dt is determined by the player's speed and direction. Since the player is moving from second to third base, x is decreasing, thus dx/dt is negative 20 feet per second.

  • What is the calculation for z in the right triangle?

    -z is calculated using the Pythagorean theorem: z^2 = x^2 + y^2. With x = 50 and y = 120, z^2 = 2500 + 14400, which gives z = √16900 ≈ 130 feet.

  • How do you find the derivative of the equation z^2 = x^2 + y^2 with respect to time?

    -The derivative with respect to time is found by differentiating each term: d(z^2)/dt = 2z * (dz/dt) and d(x^2)/dt = 2x * (dx/dt). Since y is constant, its derivative with respect to time is zero.

  • What is the final expression for dz/dt?

    -The final expression for dz/dt, after substituting the known values and simplifying, is -100/13 feet per second.

  • What is the rate at which the distance between the player and home plate is changing?

    -The rate at which the distance between the player and home plate is changing is approximately -7.69 feet per second.

  • What does the negative sign in the rate of change indicate?

    -The negative sign in the rate of change indicates that the distance between the player and home plate is decreasing as the player moves from second base towards third base.

Outlines
00:00
🏐 Baseball Diamond Problem Introduction

This paragraph introduces the baseball diamond problem, where a player is running from second base to third base at a speed of 20 feet per second. The player is 50 feet away from third base, and the baseball field is a square with side lengths of 120 feet. The goal is to determine the rate at which the distance between the player and home plate is changing. The paragraph begins by setting up the problem visually, drawing a picture of the baseball diamond and explaining the player's position relative to the bases. It then transitions into a discussion of the right triangle formed by the player's position, the third base, and home plate, and introduces the variables x, y, and z to represent the distances involved. The paragraph concludes with the formulation of the problem as finding the derivative dz/dt, which represents the rate of change of the distance to home plate.

05:01
📐 Solving the Baseball Diamond Problem

This paragraph delves into the solution of the baseball diamond problem by using the Pythagorean theorem to relate the distances x, y, and z. It explains that y, representing the distance from third base to home plate, remains constant at 120 feet, and thus its derivative with respect to time is zero. The paragraph then focuses on x, which is decreasing as the player moves from second to third base, giving dx/dt as -20 feet per second. Using the Pythagorean theorem (z^2 = x^2 + y^2), the paragraph calculates the value of z (130 feet) and then finds its derivative with respect to time. The final calculation results in a rate of change of -100/13 feet per second, or approximately -7.69 feet per second, indicating how quickly the distance between the player and home plate is decreasing as the player moves towards third base.

Mindmap
Keywords
💡baseball diamond
The term 'baseball diamond' refers to the geometric layout of a baseball field, which is shaped like a square with the bases at the corners. In the context of the video, the baseball diamond is used as the setting for the problem, where the shape and dimensions of the field are crucial to solving the mathematical problem presented.
💡player
In this context, 'player' refers to a baseball player who is running from second base to third base. The player's movement and position on the field are central to the problem being discussed, as the calculations involve determining the rate of change of distance between the player and home plate.
💡speed
Speed is a physical quantity that indicates how fast an object is moving, defined as the rate at which an object covers distance. In the video, the player's speed is given as 20 feet per second, which is used to calculate the rate of change of the player's position relative to the bases.
💡distance
Distance refers to the length of the path traveled by an object from one point to another. In the video, the distance is a key factor in determining how far the player is from the various bases and how that distance changes as the player moves.
💡rate of change
The rate of change is a mathematical concept that describes how quickly a quantity changes with respect to another quantity. In the video, the focus is on finding the rate at which the distance between the player and home plate changes as the player moves towards third base.
💡right triangle
A right triangle is a geometric figure with three right angles, one of which is at the vertex formed by the intersection of the two longest sides. In the video, the right triangle is formed by the player's position, the third base, and the home plate, which is used to apply the Pythagorean theorem to solve for unknown distances.
💡Pythagorean theorem
The Pythagorean theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the video, this theorem is used to find the distance between the player and home plate.
💡derivative
In calculus, a derivative represents the rate at which a function changes with respect to one of its variables. In the video, the derivative is used to find the rate of change of the distance between the player and home plate with respect to time.
💡constant
A constant is a value that does not change within the context of a given problem or operation. In the video, the side length of the baseball diamond field is treated as a constant because it remains the same throughout the player's movement from second to third base.
💡negative value
A negative value indicates a decrease or movement in the opposite direction. In the context of the video, a negative value for dx/dt signifies that the distance x (from the player to third base) is decreasing as the player moves towards third base.
💡decimal conversion
Decimal conversion refers to the process of expressing a number, often a fraction, in decimal form. In the video, the final answer is given as a fraction (negative 100 over 13) and then converted to a decimal for easier interpretation.
Highlights

The problem discussed is the baseball diamond problem, involving a player running from second base to third base.

The player's speed is given as 20 feet per second.

The player is 50 feet away from third base at the start of the problem.

The baseball diamond field is described as a square with side lengths of 120 feet.

The goal is to determine the rate at which the distance between the player and home plate is changing, represented as dz/dt.

A right triangle is formed in the scenario, which is instrumental in solving the problem.

The value of x (distance from second base to the player) is 50 feet.

The value of y (distance from third base to home plate) is 120 feet.

dx/dt (the rate of change of x) is negative 20 feet per second since x is decreasing.

The distance z (from the player to home plate) is calculated to be 130 feet using the Pythagorean theorem.

The derivative of the equation z^2 = x^2 + y^2 with respect to time is used to find dz/dt.

The derivative of x squared is 2x(dx/dt), and y is considered constant (120 feet) in this scenario.

The final calculated rate of change of the distance between the player and home plate is -100/13 feet per second or approximately -7.69 feet per second.

The negative sign indicates that the player is moving towards third base, thus reducing the distance to home plate.

The problem-solving approach involves a combination of geometric understanding and calculus.

The solution process is clearly explained, making it accessible for learning purposes.

Transcripts

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BaseballPhysicsMathematicsSpeedCalculationGeometrySportsScienceRateOfChangeBaseballDiamondPlayerMovementPhysicsInSportsEducationalContent