Identifying Parametric Equations in the Plane

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = 4 cos t, y = 9 sin t, 0 ≤ t ≤ 2π

검증된 단계별 안내

Start with the given parametric equations: \(x = 4 \cos t\) and \(y = 9 \sin t\), where \(0 \leq t \leq 2\pi\).

Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). We will use this to eliminate the parameter \(t\).

Express \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\): from \(x = 4 \cos t\), we get \(\cos t = \frac{x}{4}\); from \(y = 9 \sin t\), we get \(\sin t = \frac{y}{9}\).

Substitute these into the Pythagorean identity: \(\left(\frac{x}{4}\right)^2 + \left(\frac{y}{9}\right)^2 = 1\).

This equation represents the Cartesian form of the particle's path, which is an ellipse. The parameter interval \(0 \leq t \leq 2\pi\) means the particle traces the entire ellipse once, and the direction of motion corresponds to increasing \(t\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex paths and motions in the plane.

Conversion to Cartesian Equation

To identify the particle’s path, parametric equations are combined to eliminate the parameter t, resulting in a Cartesian equation relating x and y directly. This often involves using trigonometric identities or algebraic manipulation to rewrite the curve in standard form.

Graphing and Direction of Motion

Graphing the Cartesian equation shows the shape of the path, while the parameter interval and parametric equations indicate the portion traced and the direction of motion. Understanding how t changes helps to mark the start, end, and direction along the curve.