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Chapter 10: Q. 36 (page 699)

Find parametric equations for an object that moves along the ellipsex216+y29=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=4cos蟺2ty=3sin蟺2t

Step by step solution

01

Step 1. Given information:

The equation of ellipse:

x216+y29=1

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

02

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

The graph of the ellipse is:

03

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

Let,

x=4cos蝇ty=3sin蝇t, where 蝇is some constant.

These parametric equations satisfy the equation of the ellipse.

Since,

4cos蝇t216+sin蝇t29=16cos2蝇t16+9sin2蝇t9=cos2蝇t+sin2蝇t=1

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

For the motion to be counterclockwise, the motion has to begin with the value of xdecreasing and yincreasing as tincreases. This requires that 蝇>0.

Finally since 1 revolution requires 4 seconds, the period is 蝇=2蟺4蝇=蟺2.

The parametric equations that satisfy the conditions stipulated are :

x=4cos蟺2ty=3sin蟺2t

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