Chapter 11

Exer. 45-78: Sketch the graph of the polar equation. $$ \theta=-\pi / 6 $$

An arch of a bridge is semielliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 6 feet from the center of the base.

Find an equation for the set of points in an \(x y\)-plane such that the difference of the distances from \(F\) and \(F^{\prime}\) is \(k\). $$F(0,17), \quad F^{\prime}(0,-17) ; \quad k=30$$

Exer. 45-78: Sketch the graph of the polar equation. $$ \theta=\pi / 4 $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=3 \cos \theta $$

Assume that the length of the major axis of Earth's orbit is \(186,000,000\) miles and that the eccentricity is \(0.017\). Approximate, to the nearest 1000 miles, the maximum and minimum distances between Earth and the sun.

The parabola \(y^{2}=4 p(x+p)\) has its focus at the origin and axis along the \(x\)-axis. By assigning different values to \(p\), we obtain a family of confocal parabolas, as shown in the figure. Such families occur in the study of electricity and magnetism. Show that there are exactly two parabolas in the family that pass through a given point \(P\left(x_{1}, y_{1}\right)\) if \(y_{1} \neq 0\). Exercise 50

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-2 \sin \theta $$

The planet Mercury travels in an elliptical orbit that has eccentricity \(0.206\) and major axis of length \(0.774 \mathrm{AU}\). Find the maximum and minimum distances between Mercury and the sun.

A radio telescope has the shape of a paraboloid of revolution, with focal length \(p\) and diameter of base \(2 a\). From calculus, the surface area \(S\) available for collecting radio waves is $$ S=\frac{8 \pi p^{2}}{3}\left[\left(1+\frac{a^{2}}{4 p^{2}}\right)^{3 / 2}-1\right] $$ One of the largest radio telescopes, located in Jodrell Bank, Cheshire, England, has diameter 250 feet and focal length 75 feet. Approximate \(S\) to the nearest thousand square feet.

Describe the part of a hyperbola given by the equation. $$x=\frac{5}{4} \sqrt{y^{2}+16}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=4 \cos \theta+2 \sin \theta $$

A satellite will travel in a parabolic path near a planet if its velocity \(v\) in meters per second satisfies the equation \(v=\sqrt{2 k / r}\), where \(r\) is the distance in meters between the satellite and the center of the planet and \(k\) is a positive constant. The planet will be located at the focus of the parabola, and the satellite will pass by the planet once. Suppose a satellite is designed to follow a parabolic path and travel within 58,000 miles of Mars, as shown in the figure. (a) Determine an equation of the form \(x=a y^{2}\) that describes its flight path. (b) For Mars, \(k=4.28 \times 10^{13}\). Approximate the maximum velocity of the satellite. (c) Find the velocity of the satellite when its \(y\)-coordinate is 100,000 miles.

Describe the part of a hyperbola given by the equation. $$x=-\frac{5}{4} \sqrt{y^{2}+16}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=6 \cos \theta-2 \sin \theta $$

Describe the part of a hyperbola given by the equation. $$y=\frac{3}{7} \sqrt{x^{2}+49}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=4(1-\sin \theta) $$

Describe the part of a hyperbola given by the equation. $$y=-\frac{3}{7} \sqrt{x^{2}+49}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=3(1+\cos \theta) $$

Describe the part of a hyperbola given by the equation. $$y=-\frac{9}{4} \sqrt{x^{2}-16}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-6(1+\cos \theta) $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2(1+\sin \theta) $$

Describe the part of a hyperbola given by the equation. $$x=-\frac{2}{3} \sqrt{y^{2}-36}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2+4 \sin \theta $$

Describe the part of a hyperbola given by the equation. $$x=\frac{2}{3} \sqrt{y^{2}-36}$$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=1+2 \cos \theta $$

The graphs of the equations $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1 $$ are called conjugate hyperbolas. Sketch the graphs of both equations on the same coordinate plane, with \(a=5\) and \(b=3\), and describe the relationship between the two graphs.

Exer. 45-78: Sketch the graph of the polar equation. $$ r=\sqrt{3}-2 \sin \theta $$

Find an equation of the hyperbola with foci \((h \pm c, k)\) and vertices ( \(h \pm a, k\) ), where \(0

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2 \sqrt{3}-4 \cos \theta $$

Cooling tower A cooling tower, such as the one shown in the figure, is a hyperbolic structure. Suppose its base diameter is 100 meters and its smallest diameter of 48 meters occurs 84 meters from the base. If the tower is 120 meters high, approximate its diameter at the top.

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2-\cos \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=5+3 \sin \theta $$

Locating a ship A ship is traveling a course that is 100 miles from, and parallel to, a straight shoreline. The ship sends out a distress signal that is received by two Coast Guard stations A and B, located 200 miles apart, as shown in the figure. By measuring the difference in signal reception times, it is determined that the ship is 160 miles closer to \(\mathrm{B}\) than to \(\mathrm{A}\). Where is the ship?

Exer. 45-78: Sketch the graph of the polar equation. $$ r=4 \csc \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-3 \sec \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=8 \cos 3 \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2 \sin 4 \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=3 \sin 2 \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=8 \cos 5 \theta $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r^{2}=4 \cos 2 \theta \quad \text { (lemniscate) } $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2^{\theta}, \theta \geq 0 \quad \text { (spiral) } $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=e^{2 \theta}, \theta \geq 0 \quad \text { (logarithmic spiral) } $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2 \theta, \theta \geq 0 $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r \theta=1, \theta>0 \quad \text { (spiral) } $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=6 \sin ^{2}(\theta / 2) $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-4 \cos ^{2}(\theta / 2) $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2+2 \sec \theta \quad \text { (conchoid) } $$

Exer. 45-78: Sketch the graph of the polar equation. $$ r=1-\csc \theta $$

If \(P_{1}\left(r_{1}, \theta_{1}\right)\) and \(P_{2}\left(r_{2}, \theta_{2}\right)\) are points in an \(r \theta\)-plane, use the law of cosines to prove that $$ \left[d\left(P_{1}, P_{2}\right)\right]^{2}=r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right) . $$