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Chapter 10: Problem 2
Give the property that defines all ellipses.
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A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. When \(t=0,\) Earth is at (2,0) and Mars is at (3,0) both orbit the Sun (at (0,0) ) in the counterclockwise direction. The position of Mars relative to Earth is given by the parametric equations \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\) a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars relative to Earth is a lima莽on (Exercise 89).
Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?
Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=\sin 2 \theta\)
Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) \(x=a \sin ^{n} t, y=b \cos ^{n} t,\) where \(a\) and \(b\) are real numbers and \(n\) is a positive integer
The region bounded by the parabola \(y=a x^{2}\) and the horizontal line \(y=h\) is revolved about the \(y\) -axis to generate a solid bounded by a surface called a paraboloid (where \(a>0\) and \(h>0\) ). Show that the volume of the solid is \(\frac{3}{2}\) the volume of the cone with the same base and vertex.
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