Chapter 10: Q. 36 (page 699)

Find parametric equations for an object that moves along the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$
where the motion begins at $(4, 0)$ , is counterclockwise, and requires 4 seconds for a complete revolution.

Short Answer

Expert verified

The parametric equation is:

$x = 4 \cos (\frac{π}{2} t) y = 3 \sin (\frac{π}{2} t)$

Step by step solution

Step 1. Given information:

The equation of ellipse:

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Time taken by object to complete one cycle is 4 seconds.

Step 2. Draw the graph of the ellipse using a graphing calculator.

The graph of the ellipse is:

Step 3. To find the parametric equation of the ellipse.

Let,

$x = 4 \cos (ω t) y = 3 \sin (ω t)$ , where $ω$ is some constant.

These parametric equations satisfy the equation of the ellipse.

Since,

$\frac{{(4 \cos (ω t))}^{2}}{16} + \frac{{(\sin (ω t))}^{2}}{9} = \frac{16 \cos^{2} (ω t)}{16} + \frac{9 \sin^{2} (ω t)}{9} = \cos^{2} (ω t) + \sin^{2} (ω t) = 1$

Furthermore, when $t = 0$ , $x = 4$ and when $y = 0$ .

For the motion to be counterclockwise, the motion has to begin with the value of $x$ decreasing and $y$ increasing as $t$ increases. This requires that $ω > 0$ .

Finally since 1 revolution requires 4 seconds, the period is $ω = \frac{2 π}{4} ω = \frac{π}{2}$ .

The parametric equations that satisfy the conditions stipulated are :

$x = 4 \cos (\frac{π}{2} t) y = 3 \sin (\frac{π}{2} t)$

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Find parametric equations for an object that moves along the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$
where the motion begins at $(4, 0)$ , is counterclockwise, and requires 4 seconds for a complete revolution.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Draw the graph of the ellipse using a graphing calculator.

Step 3. To find the parametric equation of the ellipse.

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Find parametric equations for an object that moves along the ellipsex216+y29=1where the motion begins at 4,0, is counterclockwise, and requires 4 seconds for a complete revolution.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Draw the graph of the ellipse using a graphing calculator.

Step 3. To find the parametric equation of the ellipse.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Pure Maths

Decision Maths

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find parametric equations for an object that moves along the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$
where the motion begins at $(4, 0)$ , is counterclockwise, and requires 4 seconds for a complete revolution.