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WZ
Numerade Educator

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Problem 52 Hard Difficulty

The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations
$$ x = (2 + \cos 1.5t) \cos t $$
$$ y = (2 + \cos 1.5t) \sin t $$
$$ z = \sin 1.5t $$
to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $ xy $-plane has polar coordinates $ r = 2 + \cos 1.5t $ and $ \theta = t $, so $ r $ varies between 1 and 3. Then show that $ z $ has maximum and minimum values when the projection is halfway between $ r = 1 $ and $ r = 3 $.
When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

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