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Chapter 11: Problem 41

Find parametric equations for an object that moves along the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) with the motion described. The motion begins at \((0,3),\) is clockwise, and requires 1 second for a complete revolution.

Short Answer

Expert verified
The parametric equations are \(x(t) = 2 \cos (2 \pi t) \) and \(y(t) = -3 \sin(2 \pi t)\).

Step by step solution

01

Identify the equations of an ellipse

The standard form of an ellipse equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). From the given ellipse equation \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), identify the values of \(a\) and \(b\) where \(a^2 = 4\) and \(b^2 = 9\). Thus, \(a = 2\) and \(b = 3\).
02

Use parametric equations for the ellipse

For an ellipse centered at the origin, the parametric equations in the counterclockwise direction are: \(x(t) = a \cos(t)\) and \(y(t) = b \sin(t)\). Substituting \(a = 2\) and \(b = 3\), the equations become \(x(t) = 2 \cos(t)\) and \(y(t) = 3 \sin(t)\).
03

Account for clockwise motion

To adjust for clockwise motion, substitute \(-t\) for \(t\) to reverse the direction. The equations now become \(x(t) = 2 \cos(-t)\) and \(y(t) = 3 \sin(-t)\). Using trigonometric identities, \(\frac{\text{that} \cos(-t) = \cos(t)\ and \sin(-t) = - \sin(t)}{new step}\) = these adjust to \(x(t) = 2 \cos(t)\) and \(y(t) = - 3 \sin(t)\).
04

Set the period for 1-second revolution

Since the motion completes one revolution in 1 second, the period \(T = 1 \text{second}\). Therefore, the angle \(t\) must range from \(0\) to \(2\pi\) radians in 1 second. Adjust the parameter \(t\) by introducing a factor of \(2 \pi\): \(t = 2 \pi \theta\), where \(\theta\) ranges from \(0\) to \(1\). Hence, parametric equations depend on \(\theta\) should be: \(x(t) = 2 \cos(2 \pi \theta)\) and \(y(t) = -3 \sin(2 \pi \theta)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ellipse
An ellipse is a geometric shape that looks like an elongated circle. It is defined by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \a\: and \b\: are the ellipse's semi-major and semi-minor axes respectively. In the provided problem, the equation of the ellipse is \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). This tells us that the semi-major axis \a\: is 2 and the semi-minor axis \b\: is 3.
trigonometric identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable. In this problem, we use the identities for cosine and sine with a negative argument: \(\text{cos}(-t) = \text{cos}(t)\) and \(\text{sin}(-t) = -\text{sin}(t)\). These identities help us reverse the motion from counterclockwise to clockwise.
period of revolution
The period of revolution is the time it takes for an object to make one complete trip around a path. In this exercise, the object takes 1 second to complete a full revolution around the ellipse. The parameter \(\theta\) is adjusted by \(2 \pi\) to match this period, making \(0 \leq \theta \leq 1\) cover a full cycle in 1 second.
parametric equations
Parametric equations express the coordinates of points on a curve as functions of a parameter. In our case, x and y coordinates are functions of \(t\). For the given ellipse, the parametric equations before adjusting for direction and period are \(x(t) = 2 \text{cos}(t)\) and \(y(t) = 3 \text{sin}(t)\). After adjustments, the parametric equations become \(x(\theta) = 2 \text{cos}(2\pi\ \theta)\) and \(y(\theta) = -3 \text{sin}(2\pi\ \theta)\).

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Most popular questions from this chapter

Bill's train leaves at 8: 06 an and accelerates at the rate of 2 meters per second per second. Bill, who can run 5 meters per second, arrives at the train station 5 seconds after the train has left and runs for the train. (a) Find parametric equations that model the motions of the train and Bill as a function of time. [Hint: The position \(s\) at time \(t\) of an object having acceleration \(a\) is \(s=\frac{1}{2} a t^{2}\). (b) Determine algebraically whether Bill will catch the train. If so, when? (c) Simulate the motion of the train and Bill by simultaneously graphing the equations found in part (a).

A rectangle is inscribed in an ellipse with major axis of length 14 meters and minor axis of length 4 meters. Find the maximum area of a rectangle inscribed in the ellipse. Round your answer to two decimal places.

Use a graphing utility to graph the plane curve defined by the given parametric equations. \(x(t)=4 \sin t-2 \sin (2 t)\) \(y(t)=4 \cos t-2 \cos (2 t)\)

Problems \(67-75\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Graph the equation \(3 x-4 y=8\) on the \(x y\) -plane.

Sean throws a baseball with an initial speed of 145 feet per second at an angle of \(20^{\circ}\) to the horizontal. The ball leaves Sean's hand at a height of 5 feet. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) Determine the horizontal distance that the ball travels. (d) When is the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a).
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