PreCalc H ­ Aim #75 Word Problems.notebook
Aim #75: How do we solve word problems using conic sections?
Do Now:
The eccentricity of an ellipse and a hyperbola are both given as
Determine the following:
a. if e is greater or less than 1 in an ellipse
b. if e is greater or less than 1 in a hyperbola
c. what happens to the branches of a hyperbola if c is close to
or far away from a?
eg. The main cables of a suspension bridge are 20 meters above
the road at the towers and 4 meters above the road at the
center. The road is 80 meters long. Vertical cables are spaced
every 10 meters. The main cables hang in the shape of a parabola.
Find the equation of the parabola. Then, determine how high the
main cable is 20 meters from the center.
1
PreCalc H ­ Aim #75 Word Problems.notebook
eg. In a whispering gallery, a person standing at one focus of an
ellipse can be heard by another person standing at the other
focus, because all the sound waves that reach the ceiling from
one focus are reflected to the other focus. Where are the foci
located?
Wrap it Up!
According to Kepler‛s Laws, planets have elliptical orbits, with the
sun at one of the foci. The farthest Pluto gets from the sun is 7.4
billion kilometers. The closest it gets to the sun is 4.4 billion
kilometers. Find the equation of Pluto‛s orbit assuming a center at
(0,0).
2
PreCalc H ­ Aim #75 Word Problems.notebook
An elliptically shaped garden is surrounded by a wood walkway.
The garden is 15 meters long and 8 meters wide. The walkway is 2
meters wide. Find the equation describing the ellipse that includes
both the garden and the walkway.
The Statuary Hall in the United States Capitol is elliptical. It
measures 46 feet wide and 96 feet long. If a person is standing at
one focus, her whisper can be heard by a person standing at the
other focus. How far apart are the two people?
3
PreCalc H ­ Aim #75 Word Problems.notebook
An arch of a bridge over a highway is semi-elliptical in shape and
42 ft. across. The highest point of the arch is 14 feet above the
highway. What is the maximum height, to the nearest inch, of a
truck 8 ft. wide that can fit under the arch? (Assume the
highway is one lane)
Two radar sites are tracking an airplane that is flying on a
hyperbolic path. The first radar site is located at (0, 0), and
shows the airplane to be 200 meters away at a certain time. The
second radar site, located 160 miles east of the first, shows the
airplane to be 100 meters away at this same time. Find the
coordinates of all possible points where the airplane could be
located. (Find the equation of the hyperbola where the plane could
be located)
4
PreCalc H ­ Aim #75 Word Problems.notebook
Alpha particles are deflected along hyperbolic paths when they are
directed towards the nuclei of gold atoms. If an alpha particle
gets as close as 10 units to the nucleus along a hyperbolic path
with asymptote y=(2/5)x, what is the equation of its path?
5