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A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.
0.5625 feet
Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.
A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and the height at the center of the gallery?
Length is 96 feet and height is approximately 26.53 feet.
For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units (AU).
Halley’s Comet: length of major axis = 35.88, eccentricity = 0.967
Hale-Bopp Comet: length of major axis = 525.91, eccentricity = 0.995
$r=\frac{2.616}{1+0.995\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}$
Mars: length of major axis = 3.049, eccentricity = 0.0934
Jupiter: length of major axis = 10.408, eccentricity = 0.0484
$r=\frac{5.192}{1+0.0484\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}$
True or False? Justify your answer with a proof or a counterexample.
The rectangular coordinates of the point $\left(4,\frac{5\pi}{6}\right)$ are $\left(2\sqrt{3},\mathrm{-2}\right).$
The equations $x=\text{cosh}(3t),$ $y=2\phantom{\rule{0.2em}{0ex}}\text{sinh}(3t)$ represent a hyperbola.
True.
The arc length of the spiral given by $r=\frac{\theta}{2}$ for $0\le \theta \le 3\pi $ is $\frac{9}{4}{\pi}^{3}.$
Given $x=f(t)$ and $y=g(t),$ if $\frac{dx}{dy}=\frac{dy}{dx},$ then $f(t)=g(t)+\text{C,}$ where C is a constant.
False. Imagine $y=t+1,$ $x=\text{\u2212}t+1.$
For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
$x=1+t,$ $y={t}^{2}-1,$ $\mathrm{-1}\le t\le 1$
$x=\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=1-\text{csc}\phantom{\rule{0.2em}{0ex}}\theta ,$ $0\le \theta \le 2\pi $
$x=4\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\varphi ,$ $y=1-\text{sin}\phantom{\rule{0.2em}{0ex}}\varphi ,$ $0\le \varphi \le 2\pi $
$\frac{{x}^{2}}{16}+{(y-1)}^{2}=1$
For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any.
$r=4\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{\theta}{3}\right)$
$r=5\phantom{\rule{0.2em}{0ex}}\text{cos}\left(5\theta \right)$
Symmetric about polar axis
For the following exercises, find the polar equation for the curve given as a Cartesian equation.
${y}^{2}=4+{x}^{2}$
${r}^{2}=\frac{4}{{\text{sin}}^{2}\theta -{\text{cos}}^{2}\theta}$
For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line.
$x=\text{ln}(t),$ $y={t}^{2}-1,$ $t=1$
$r=3+\text{cos}\left(2\theta \right),$ $\theta =\frac{3\pi}{4}$
$y=\frac{3\sqrt{2}}{2}+\frac{1}{5}\left(x+\frac{3\sqrt{2}}{2}\right)$
Find $\frac{dy}{dx},$ $\frac{dx}{dy},$ and $\frac{{d}^{2}x}{d{y}^{2}}$ of $y=\left(2+{e}^{\text{\u2212}t}\right),$ $x=1-\text{sin}(t)$
For the following exercises, find the area of the region.
$x={t}^{2},$ $y=\text{ln}(t),$ $0\le t\le e$
$\frac{{e}^{2}}{2}$
$r=1-\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ in the first quadrant
For the following exercises, find the arc length of the curve over the given interval.
$r=6\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $0\le \theta \le 2\pi .$ Check your answer by geometry.
For the following exercises, find the Cartesian equation describing the given shapes.
A parabola with focus $(2,\mathrm{-5})$ and directrix $x=6$
${\left(y+5\right)}^{2}=\mathrm{-8}x+32$
An ellipse with a major axis length of 10 and foci at $\left(\mathrm{-7},2\right)$ and $\left(1,2\right)$
A hyperbola with vertices at $(3,\mathrm{-2})$ and $(\mathrm{-5},\mathrm{-2})$ and foci at $(\mathrm{-2},\mathrm{-6})$ and $(\mathrm{-2},4)$
$\frac{{\left(y+1\right)}^{2}}{16}-\frac{{\left(x+2\right)}^{2}}{9}=1$
For the following exercises, determine the eccentricity and identify the conic. Sketch the conic.
$r=\frac{6}{1+3\phantom{\rule{0.2em}{0ex}}\text{cos}(\theta )}$
$r=\frac{4}{3-2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}$
$e=\frac{2}{3},$ ellipse
$r=\frac{7}{5-5\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}$
Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?
$\frac{{y}^{2}}{{19.03}^{2}}+\frac{{x}^{2}}{{19.63}^{2}}=1,$ $e=0.2447$
The C/1980 E1 comet was observed in 1980. Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364 AU, find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? ( Hint : Consider the Sun at point $(0,0).)$
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