Math 2312 Precalculus
Lab 3
Professor Merrill
Name ________________________________
Show work and Circle answers
Solve the following systems by the most convenient method:
1.
2.
3. The perimeter of a rectangular concrete slab is 98 feet and its area is 444 square feet.
What is the length of the longer side of the slab?
4. Sue bought some saltwater fish for $2 each and some freshwater fish for $4 each. If she
bought a total of 17 fish and spent a total of $56, how many freshwater fish did she buy?
5. The opening under a bridge is in the shape of a semielliptical arch and has a span of 82 feet.
The height of the arch, at a distance of 9 feet from the center, is 30 feet. Find the height of
the arch at its center.
6. Write an equation of the ellipse whose
vertices are (0, -13), (0, 13) and whose
co-vertices are (-12, 0), (12, 0). Find the
foci.
7. Write the equation of the
ellipse shown below.
8. A satellite orbits around the moon in an elliptical path. Assuming the
moon to be at the center of a rectangular coordinate system, find the
equation of the elliptical path of a satellite whose x-intercepts are
±82,000km and whose y-intercepts are ±71,000km.
9. The US Capital building contains an elliptical room. It is 96ft in length
and 46ft in width. Write an equation to describe the shape of the room.
Assume that it is centered at the origin and the major axis is
horizontal. John Quincy Adams discovered that he could overhear
conversations being held at the opposing party leader’s desk if he stood
in a certain spot in the elliptical chamber. Describe the position of the
desk and how far away Adams had to stand to overhear.
10. Suppose that you are Chief Mathematician for Ornery and Sly
Construction Company. Your company has a contract to build a football
stadium in the form of two concentric ellipses, with the field inside the
inner ellipse, and the seats between the two ellipses. The seats are in
the intersection of the graphs of
X2 + 4y2  100
and
25x2 + 36y2  3600.
where each unit on the graph represents 10 meters.
a.
b.
Draw a graph of the seating area.
The area of an elliptical region is ab, where a and b are the semimajor/minor-axis respectively, and   3.14159. The Engineering
Department estimates that each seat occupies 0.8 square meters. What is
the seating capacity of the stadium?
11. Graph and give the following information:
4 x 2  9( y  1)2  36
transverse axis:
center:
vertices:
foci:
asymptotes:
12. Suppose that you have been hired by the Palomar Observatory near San Diego. Your
assignment is to track incoming meteorites to find out whether or not they will strike
the Earth. Since the Earth has a circular cross-section, you decide to set up a
Cartesian Coordinate System with its origin at the center of the Earth. The equation of
the Earth's surface is
x2 + y2 = 40,
where x and y are distances in thousands of kilometers.
a.
The first meteorite you observe is moving along the parabola whose equation is
18x - y2 = -144.
Will this meteorite strike the Earth's surface? If so, where? If not, how do you
tell?
b.
The second meteorite is coming in from the lower left along one branch of the
hyperbola
4x2 - y2 - 80x = -340
Will it strike the Earth's surface? If so, where (ordered pair)? If not, how do
you tell?
c.
To the nearest 100 kilometers, what is the radius of the Earth?
13. In each of the following give the coordinates of the vertex and focus, the equation
of the directrix, and then graph the parabola.
a) ( y  4)2  12  x  2
b) ( x  1)2  8 y
vertex:
vertex:
focus:
focus:
directrix:
directrix:
14. Sunfire is a glass parabola used to collect solar energy. The sun’s rays are
reflected from the mirrors toward two boilers located at the focus of the parabola.
When heated, the boilers produce steam that powers an alternator to produce
electricity.
a) Write an equation for Sunfire’s cross section.
b) How deep is the dish?
15. The towers of the Golden Gate Bridge connecting San Francisco to Marin County are
1280 meters apart and rise 160 meters above the road. The cable between the towers
has the shape of a parabola, and the cable just touches the sides of the road, midway
between the towers. What is the height of the cable 200 meters from a tower?
160 meters
1280 meters
Write the partial fraction decomposition of the following rational expressions:
x 2
2x  3
16.
17.
x (x  4)
(x  1) 2
18.
x 3  2x 2  x  1
x 2  3x  4
19.
3x 2  7x  2
x3 x