Maximum Area Triangle Curve Y=8sin((x-π)/3) And Line Y=4

by ADMIN 57 views

Introduction

This article delves into a fascinating problem involving trigonometric functions and geometry. We aim to determine the maximum area of triangle PAB, where points A and B are the intersections of the curve y = 8sin((x - π)/3) (for 0 ≤ x ≤ 10π) and the line y = 4, and P is a point on the curve (excluding points on the line y = 4). This problem combines trigonometric equations, graphical analysis, and geometric principles, making it an excellent exercise in mathematical reasoning and problem-solving.

Understanding the Curve y = 8sin((x - π)/3)

Before we can tackle the main problem, we need to thoroughly understand the behavior of the curve y = 8sin((x - π)/3). This is a sinusoidal function, specifically a sine wave, that has undergone several transformations. Let's break down these transformations step by step:

  1. Amplitude: The coefficient 8 in front of the sine function stretches the graph vertically. The amplitude of this wave is 8, meaning the maximum value of y is 8 and the minimum value is -8.
  2. Horizontal Shift: The term (x - π) inside the sine function represents a horizontal shift. Specifically, it shifts the graph π units to the right. This means the standard sine wave, which normally starts at (0, 0), now starts at (π, 0).
  3. Horizontal Compression/Stretch: The term (x - π)/3 affects the period of the sine wave. The period of the standard sine wave y = sin(x) is 2π. Dividing the argument of the sine function by 3 stretches the graph horizontally, making the period 3 times longer. Therefore, the period of y = 8sin((x - π)/3) is 3 * 2π = 6π.

Considering the domain 0 ≤ x ≤ 10π, we need to determine how many full periods of the sine wave are included. Since the period is 6π, there is one full period within 0 to 6π, and a fraction of another period within 6π to 10π. This means the graph will complete one full cycle and part of a second cycle within the given domain. Understanding these transformations is crucial for visualizing the curve and identifying the points of intersection with the line y = 4.

Finding the Intersection Points A and B

To determine the points A and B, we need to solve the equation 8sin((x - π)/3) = 4. This equation represents the x-values where the curve intersects the line y = 4. Let's solve it step-by-step:

  1. Divide by 8: Divide both sides of the equation by 8 to isolate the sine function: sin((x - π)/3) = 1/2.
  2. Find the General Solutions: We know that sin(θ) = 1/2 for θ = π/6 and θ = 5π/6 in the interval [0, 2π]. Therefore, the general solutions for θ are: θ = π/6 + 2πk and θ = 5π/6 + 2πk, where k is an integer.
  3. Substitute and Solve for x: Now, substitute θ = (x - π)/3 and solve for x:
    • (x - π)/3 = π/6 + 2πk
      • Multiply both sides by 3: x - π = π/2 + 6πk
      • Add π to both sides: x = 3π/2 + 6πk
    • (x - π)/3 = 5π/6 + 2πk
      • Multiply both sides by 3: x - π = 5π/2 + 6πk
      • Add π to both sides: x = 7π/2 + 6πk
  4. Find Solutions in the Domain 0 ≤ x ≤ 10π: We need to find the values of k that give us solutions for x within the given domain. Let's analyze each case:
    • x = 3π/2 + 6πk
      • For k = 0: x = 3π/2 (This is within the domain)
      • For k = 1: x = 3π/2 + 6π = 15π/2 (This is within the domain)
    • x = 7π/2 + 6πk
      • For k = 0: x = 7π/2 (This is within the domain)

Thus, the x-coordinates of the intersection points are 3π/2, 7π/2, and 15π/2. The corresponding y-coordinate for all these points is 4 (since they lie on the line y = 4). Therefore, we can define the points A and B as follows: Let A = (3π/2, 4) and B = (7π/2, 4). The order is not important at this stage.

Maximizing the Area of Triangle PAB

Now comes the crux of the problem: finding the point P on the curve that maximizes the area of triangle PAB. The area of a triangle can be calculated using the formula:

Area = (1/2) * base * height

In this case, we can consider the line segment AB as the base of the triangle. The length of the base AB is the distance between points A and B, which is:

AB = |7π/2 - 3π/2| = 2π

To maximize the area, we need to maximize the height of the triangle. The height is the perpendicular distance from point P to the line AB (which is the line y = 4). Since P lies on the curve y = 8sin((x - π)/3), the height of the triangle is the difference between the y-coordinate of P and the y-coordinate of the line AB (which is 4):

Height = |8sin((x - π)/3) - 4|

To maximize the height, we need to find the maximum value of the expression |8sin((x - π)/3) - 4|. This occurs when the sine function is at its maximum or minimum value. The maximum value of sin((x - π)/3) is 1, and the minimum value is -1. Let's consider both cases:

  1. When sin((x - π)/3) = 1:
    • Height = |8(1) - 4| = 4
  2. When sin((x - π)/3) = -1:
    • Height = |8(-1) - 4| = 12

Clearly, the maximum height occurs when sin((x - π)/3) = -1, and the maximum height is 12.

Calculating the Maximum Area

Now that we have the maximum height and the base length, we can calculate the maximum area of triangle PAB:

Maximum Area = (1/2) * base * maximum height = (1/2) * 2π * 12 = 12π

Determining the Point P

To completely solve the problem, let's find the x-coordinate of the point P where the maximum height occurs. We know that sin((x - π)/3) = -1. The general solution for this equation is:

(x - π)/3 = 3π/2 + 2πk, where k is an integer.

Solving for x:

  • x - π = 9π/2 + 6πk
  • x = 11π/2 + 6πk

We need to find a value of k that gives us a solution within the domain 0 ≤ x ≤ 10π. For k = 0, we get x = 11π/2, which is within the domain. Therefore, one possible point P is (11π/2, -8). Other solutions for x exist for different values of k, but they will result in the same maximum height and area.

Conclusion

In conclusion, the maximum area of triangle PAB, where A and B are the intersection points of the curve y = 8sin((x - π)/3) and the line y = 4, and P is a point on the curve (not on the line), is 12π. This problem beautifully illustrates the interplay between trigonometry and geometry, requiring a solid understanding of sinusoidal functions, equation solving, and geometric principles. By carefully analyzing the curve, finding the intersection points, and maximizing the height of the triangle, we successfully determined the maximum area. This exercise highlights the power of mathematical reasoning and problem-solving in tackling complex challenges.

Summary of Key Concepts

  • Transformations of Trigonometric Functions: Understanding amplitude, horizontal shifts, and period changes in sinusoidal functions is crucial for graphing and analyzing trigonometric curves.
  • Solving Trigonometric Equations: The ability to find general solutions for trigonometric equations is essential for determining intersection points and other critical values.
  • Geometric Principles: The formula for the area of a triangle and the concept of perpendicular distance are fundamental to solving geometric optimization problems.
  • Optimization Techniques: Maximizing the area of the triangle involves identifying the critical points of the height function and using calculus or geometric reasoning to find the maximum value.

This detailed exploration provides a comprehensive understanding of the problem and its solution, emphasizing the interconnectedness of various mathematical concepts.

Practice Problems

To further solidify your understanding, try solving similar problems with different curves and lines. For instance, you could explore:

  1. Finding the maximum area of a triangle formed by the intersection of a cosine curve and a horizontal line.
  2. Determining the maximum area of a triangle formed by the intersection of a trigonometric curve and a diagonal line.
  3. Investigating how changes in the amplitude, period, or phase shift of the trigonometric curve affect the maximum area of the triangle.

By tackling these practice problems, you can enhance your problem-solving skills and deepen your appreciation for the beauty and power of mathematics.

Further Exploration

For those interested in delving deeper into this topic, consider exploring the following areas:

  • Calculus Applications: Use calculus techniques, such as finding derivatives and critical points, to optimize the area of the triangle.
  • Parametric Equations: Represent the curve using parametric equations and explore how this representation simplifies the problem.
  • Geometric Software: Utilize geometric software packages to visualize the curve, the line, and the triangle, and to experiment with different point positions.

By pursuing these avenues of exploration, you can expand your mathematical horizons and gain a more profound understanding of the interplay between trigonometry, geometry, and calculus.