# 7.5 Conic sections  (Page 11/23)

 Page 11 / 23

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

$r=\frac{5.192}{1+0.0484\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta }$

## Chapter review exercises

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 $\left(2\sqrt{3},-2\right).$

The equations $x=\text{cosh}\left(3t\right),$ $y=2\phantom{\rule{0.2em}{0ex}}\text{sinh}\left(3t\right)$ 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 $x=f\left(t\right)$ and $y=g\left(t\right),$ if $\frac{dx}{dy}=\frac{dy}{dx},$ then $f\left(t\right)=g\left(t\right)+\text{C,}$ where C is a constant.

False. Imagine $y=t+1,$ $x=\text{−}t+1.$

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

$x=1+t,$ $y={t}^{2}-1,$ $-1\le t\le 1$

$x={e}^{t},$ $y=1-{e}^{3t},$ $0\le t\le 1$

$y=1-{x}^{3}$

$x=\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=1-\text{csc}\phantom{\rule{0.2em}{0ex}}\theta ,$ $0\le \theta \le 2\pi$

$x=4\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\varphi ,$ $y=1-\text{sin}\phantom{\rule{0.2em}{0ex}}\varphi ,$ $0\le \varphi \le 2\pi$

$\frac{{x}^{2}}{16}+{\left(y-1\right)}^{2}=1$

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any.

$r=4\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{\theta }{3}\right)$

$r=5\phantom{\rule{0.2em}{0ex}}\text{cos}\left(5\theta \right)$

For the following exercises, find the polar equation for the curve given as a Cartesian equation.

$x+y=5$

${y}^{2}=4+{x}^{2}$

${r}^{2}=\frac{4}{{\text{sin}}^{2}\theta -{\text{cos}}^{2}\theta }$

For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line.

$x=\text{ln}\left(t\right),$ $y={t}^{2}-1,$ $t=1$

$r=3+\text{cos}\left(2\theta \right),$ $\theta =\frac{3\pi }{4}$

$y=\frac{3\sqrt{2}}{2}+\frac{1}{5}\left(x+\frac{3\sqrt{2}}{2}\right)$

Find $\frac{dy}{dx},$ $\frac{dx}{dy},$ and $\frac{{d}^{2}x}{d{y}^{2}}$ of $y=\left(2+{e}^{\text{−}t}\right),$ $x=1-\text{sin}\left(t\right)$

For the following exercises, find the area of the region.

$x={t}^{2},$ $y=\text{ln}\left(t\right),$ $0\le t\le e$

$\frac{{e}^{2}}{2}$

$r=1-\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$ in the first quadrant

For the following exercises, find the arc length of the curve over the given interval.

$x=3t+4,$ $y=9t-2,$ $0\le t\le 3$

$9\sqrt{10}$

$r=6\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $0\le \theta \le 2\pi .$ Check your answer by geometry.

For the following exercises, find the Cartesian equation describing the given shapes.

A parabola with focus $\left(2,-5\right)$ and directrix $x=6$

${\left(y+5\right)}^{2}=-8x+32$

An ellipse with a major axis length of 10 and foci at $\left(-7,2\right)$ and $\left(1,2\right)$

A hyperbola with vertices at $\left(3,-2\right)$ and $\left(-5,-2\right)$ and foci at $\left(-2,-6\right)$ and $\left(-2,4\right)$

$\frac{{\left(y+1\right)}^{2}}{16}-\frac{{\left(x+2\right)}^{2}}{9}=1$

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic.

$r=\frac{6}{1+3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\theta \right)}$

$r=\frac{4}{3-2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta }$

$e=\frac{2}{3},$ 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 $\left(0,0\right).\right)$

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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