Chapter 10

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.

Sketch the graph of the polar equation. $$r=-2 \sin \theta$$

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

Exer. \(67-68\) : The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity \(e,\) graph the orbit for the planet. Center the major axis on the \(x\) -axis, and plot the location of the sum at one focus. $$\text { Earth's path } a=149.6, e=0.093$$

Sketch the graph of the polar equation. $$r=4 \cos \theta+2 \sin \theta$$

Graph the equation. $$x=-y^{2}+2 y+5$$

Exer. \(67-68\) : The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity \(e,\) graph the orbit for the planet. Center the major axis on the \(x\) -axis, and plot the location of the sum at one focus. $$\text { Pluto's path } a=5913, \quad e=0.249$$

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

Graph the equation. $$x=2 y^{2}+3 y-7$$

Exer. \(69-72:\) Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{x^{2}}{2.9}+\frac{y^{2}}{2.1}=1 ; \quad \frac{x^{2}}{4.3}+\frac{(y-2.1)^{2}}{4.9}=1$$

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

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

Graph the parabolas on the same coordinate plane, and estimate the points of intersection. $$y=x^{2}-2.1 x-1 ; \quad x=y^{2}+1$$

Exer. \(69-72\) : Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{x^{2}}{3.9}+\frac{y^{2}}{2.4}=1 ; \quad \frac{(x+1.9)^{2}}{4.1}+\frac{y^{2}}{2.5}=1$$

Sketch the graph of the polar equation. $$r=3(1+\cos \theta)$$

Graph the parabolas on the same coordinate plane, and estimate the points of intersection. $$y=-2.1 x^{2}+0.1 x+1.2 ; \quad x=0.6 y^{2}+1.7 y-1.1$$

Exer. \(69-72\) : Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{(x+0.1)^{2}}{1.7}+\frac{y^{2}}{0.9}=1 ; \quad \frac{x^{2}}{0.9}+\frac{(y-0.25)^{2}}{1.8}=1$$

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

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

Exer. \(69-72\) : Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{x^{2}}{3.1}+\frac{(y-0.2)^{2}}{2.8}=1 ; \quad \frac{(x+0.23)^{2}}{1.8}+\frac{y^{2}}{4.2}=1$$

Sketch the graph of the polar equation. $$r=2(1+\sin \theta)$$

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

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.

Sketch the graph of the polar equation. $$r=1+2 \cos \theta$$

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

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. GRAPH CANT COPY

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

Airplane maneuver An airplane is flying along the hyperbolic path illustrated in the figure. If an equation of the path is \(2 y^{2}-x^{2}=8,\) determine how close the airplane comes to a town located at \((3,0) .\) (Hint: Let \(S\) denote the square of the distance from a point \((x, y)\) on the path to \((3,0),\) and find the minimum value of \(S\).) PICTURE CANT COPY

Sketch the graph of the polar equation. $$r=2-\cos \theta$$

Sketch the graph of the polar equation. $$r=5+3 \sin \theta$$

Sketch the graph of the polar equation. $$r=4 \csc \theta$$

Graph the hyperbolas on the same coordinate plane, and estimate their first- quadrant point of intersection.. $$\begin{aligned}&\frac{(y-0.1)^{2}}{1.6}-\frac{(x+0.2)^{2}}{0.5}=1\\\&\frac{(y-0.5)^{2}}{2.7}-\frac{(x-0.1)^{2}}{5.3}=1\end{aligned}$$

Sketch the graph of the polar equation. $$r=-3 \sec \theta$$

Graph the hyperbolas on the same coordinate plane, and estimate their first- quadrant point of intersection.. $$\frac{(x-0.1)^{2}}{0.12}-\frac{y^{2}}{0.1}=1 ; \frac{x^{2}}{0.9}-\frac{(y-0.3)^{2}}{2.1}=1$$

Sketch the graph of the polar equation. $$r=8 \cos 3 \theta$$

Graph the hyperbolas on the same coordinate plane, and determine the number of points of intersection. $$\frac{(x-0.3)^{2}}{1.3}-\frac{y^{2}}{2.7}=1 ; \frac{y^{2}}{2.8}-\frac{(x-0.2)^{2}}{1.2}=1$$

Sketch the graph of the polar equation. $$r=2 \sin 4 \theta$$

Graph the hyperbolas on the same coordinate plane, and determine the number of points of intersection. $$\begin{aligned} &\frac{(x+0.2)^{2}}{1.75}-\frac{(y-0.5)^{2}}{1.6}=1\\\&\frac{(x-0.6)^{2}}{2.2}-\frac{(y+0.4)^{2}}{2.35}=1\end{aligned}$$

Sketch the graph of the polar equation. $$r=3 \sin 2 \theta$$

Sketch the graph of the polar equation. $$r=8 \cos 5 \theta$$

Sketch the graph of the polar equation. \(r^{2}=4 \cos 2 \theta\) (lemniscate)

Sketch the graph of the polar equation. \(r=e^{20} \cdot \theta \geq 0\) (logarithmic spiral)

Sketch the graph of the polar equation. $$r=2 \theta, \theta \geq 0$$

Sketch the graph of the polar equation. \(r \theta=1, \theta>0\) (spiral)

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

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

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 $$\left[d\left(P_{1}, P_{2}\right)\right]^{2}=r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)$$

Sketch the graph of the polar equation. Prove that the graph of each polar equation is a circle, and find its center and radius. (a) \(r=a \sin \theta, a \neq 0\) (b) \(r=b \cos \theta, b \neq 0\) (c) \(r=a \sin \theta+b \cos \theta, a \neq 0\) and \(b \neq 0\)

Suppose that a radio station has two broadcasting towers located along a north-south line and that the towers are separated by a distance of \(\frac{1}{2} \lambda,\) where \(\lambda\) is the wavelength of the station's broadcasting signal. Then the intensity \(I\) of the signal in the direction \(\theta\) can be expressed by the given equation, where \(I_{0}\) is the maximum intensity of the signal. (a) Plot \(I\) using polar coordinates with \(I_{0}=5\) for \(\boldsymbol{\theta} \in \mathbf{[ 0 , 2 \pi ]}\) (b) Determine the directions in which the radio signal has maximum and minimum intensity. $$I=\frac{1}{2} I_{0}[1+\cos (\pi \sin \theta)]$$