Pre-Calculus
Unit a ttl51 i -4: Ellipse Applications
Name:
Use the data in the table to answer the next 2 questions:
anet
Semimajor
Axis (Gm)
•
Eccentricity
57.9
108.2
149.6
227.9
778.3
1427
2869
4497
5900
0.2056
0.0068
0.0167
0.0934
0.0485
0.0560
0.0461
0.0050
0.2484
tercury
snus
arth
lars
apiter
aturn
Iranus
Neptune
>luto
Source: Shupe, et ai, National Geographic Atlas of the
World (rev. 6th ed.j. Washington, JDC.1 National
0~""'-onMrr;Society, 1992, plate 116, and other sources.
1 . Find the perihelion and aphelion distances of Saturn.
/ro6,
2. Given the diameter of the Sun is about 1.392 Gm, how close does Mercury get to the Sun's
surface?
3. The orbit of Halley's comet is 36.18 Astronomical Units (AU) long and 9.12 AU wide. What is
its eccentricity?
4. The designer is building a new sound room elliptical in shape. If the major axis in 200 feet and
the minor axis is 80 feet. The designer wants to place recording speakers at the best locations on
the major axis to get the best sound reception. How far apart should the speakers be placed?
5. Halley's Comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is
approximately 0.967. The length of the major axis of the orbit is approximately 35.88 astronomical
units. (An astronomical unit is about 93 million miles.)
a. Find an equation of the orbit. Place the center of the orbit at the origin, and place the
major axis on the x-axis.
X2
b. Find the aphelion (greatest) and perihelion (smallest) distances from the sun's center to
the comet's center.
6. A hall 100 feet in length is to be designed as a whispering gallery. If the foci are
located 25 feet from the center, how high will the ceiling be at the center?
7. Mars is in an elliptical orbit around the Sun, with the Sun at one focus. The aphelion
(the point farthest from the sun) and the perihelion (the point closest to the Sun) are 155
million miles and 128 million miles, respectively, as shown in the figure below.
Mars orbit
Aphelion
Perihelion
a. How long is the major axis of the ellipse?
b. Find the focal radius, "c" and minor radius, "b"
C -
c. Write a Cartesian (rectangular)equation for the orbit with the center at the origin and the
major axis along the x axis.
X
d. At the two equinoxes (times of equal day and night), the angle at the Sun between the
major axis and Mars is 90 degrees. At these times how far is Mars from the Sun?