Problem 70 - Thomas Calculus

Question

Exercises \(67-78\) give parametric equations and parameter intervals for the motion of a particle in the \(x y\) -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$ x=4 \sin t, \quad y=5 \cos t, \quad 0 \leq t \leq 2 \pi $$

Step-by-Step Solution

Answer

The particle traces an ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\), moving counterclockwise from (0, 5).

Step 1: Identify the Parametric Equations

The given parametric equations are \( x = 4 \sin t \) and \( y = 5 \cos t \) for the parameter \( t \) such that \( 0 \leq t \leq 2\pi \). These equations describe the motion of a particle in the plane.

Step 2: Eliminate the Parameter

To find a Cartesian equation, eliminate the parameter \( t \). Recall the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). Solve each parametric equation for \( \sin t \) and \( \cos t \):- \( \sin t = \frac{x}{4} \)- \( \cos t = \frac{y}{5} \)Substitute these into the identity: \( \left(\frac{x}{4}\right)^2 + \left(\frac{y}{5}\right)^2 = 1 \).

Step 3: Simplify the Cartesian Equation

Simplify the equation \( \left(\frac{x}{4}\right)^2 + \left(\frac{y}{5}\right)^2 = 1 \) to obtain a Cartesian form. The equation becomes: \(\frac{x^2}{16} + \frac{y^2}{25} = 1\). This is the equation of an ellipse.

Step 4: Analyze and Graph the Cartesian Equation

The ellipse \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \) has its center at the origin, with a semi-major axis length of 5 (along the y-direction) and a semi-minor axis length of 4 (along the x-direction). Plot this ellipse on the Cartesian plane.

Step 5: Determine the Path and Direction

As \( t \) varies from 0 to \( 2\pi \), \( x \) and \( y \) trace the path of an ellipse starting at the point (0, 5) when \( t = 0 \). As \( t \) increases, the particle moves counterclockwise around the ellipse, completing one full revolution when \( t = 2\pi \).

Key Concepts

Cartesian Equation

A Cartesian equation is a mathematical representation that relates two variables, typically denoted as \(x\) and \(y\), without involving any parameters. In the context of this exercise, we start with parametric equations \(x = 4 \sin t\) and \(y = 5 \cos t\).
To find the Cartesian equation, we eliminate the parameter \(t\). By using trigonometric identities and substitutions, we derive the equation from parametric to Cartesian form. Here, we apply the Pythagorean Identity and transform it into the Cartesian equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
This expression, which no longer depends on the parameter \(t\), describes the relationship between \(x\) and \(y\) as part of an ellipse on a Cartesian plane.

Pythagorean Identity

The Pythagorean Identity is a fundamental trigonometric identity given by \(\sin^2 t + \cos^2 t = 1\).
This identity is instrumental when converting parametric equations into Cartesian equations.
In this exercise, the identity is used to connect the expressions derived from \(x = 4 \sin t\) and \(y = 5 \cos t\). We solve these equations for \(\sin t\) and \(\cos t\), respectively:

• \(\sin t = \frac{x}{4}\)
• \(\cos t = \frac{y}{5}\)

Substituting into the Pythagorean Identity, \((\frac{x}{4})^2 + (\frac{y}{5})^2 = 1\) allows us to derive a Cartesian equation. This use of the identity helps bridge parametric equations into a form that is more easily studied, usually as a type of conic section.

Ellipse

An ellipse is a type of conic section that appears as an elongated circle. Its mathematical representation is, in this case, derived into the equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).
This form describes an ellipse centered at the origin \((0, 0)\) on the Cartesian plane.
The equation of an ellipse generally appears as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a\) and \(b\) represent the semi-minor and semi-major axes, respectively.

• Here, \(a = 4\) and \(b = 5\) indicate the lengths of these axes.

The parameter interval \(0 \leq t \leq 2 \pi\) outlines the complete traversal of the ellipse, describing the path of a particle moving in this shape.

Particle Motion in the Plane

Particle motion in the plane often involves describing the path and movement of a particle using parametric equations. In physics and mathematics, these equations are a way to define the coordinates \((x, y)\) over time.
For this exercise, the particle's motion described by \(x = 4\sin t\) and \(y = 5\cos t\) outlines a trajectory on a 2D plane as \(t\) spans from \(0\) to \(2\pi\). This interval covers one full rotation around the ellipse.

• The motion starts at \((0, 5)\) when \(t = 0\).
• As \(t\) increases, the particle moves counterclockwise.
• At \(t = \frac{\pi}{2}\), it reaches \((4, 0)\) and continues the elliptical path.

Understanding particle motion can aid in physics and other applied sciences where trajectory and speed variations are studied.