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Math 2312 Precalculus

Lab 3

Professor Merrill

Name ________________________________

Solve the following systems by the most convenient method:

1.

2.

**Show work and****Circle answers**

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

7. Write the equation of the ellipse shown below. co-vertices are (-12, 0), (12, 0). Find the foci.

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

X

2

+ 4y

2

100 and 25x

2

+ 36y

2

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?

c.

b.

a.

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 x 2 + y 2 = 40, where *x* and *y* are distances in *thousands *of kilometers.

The first meteorite you observe is moving along the parabola whose equation is

18x - y

2

= -144.

Will this meteorite strike the Earth's surface? If so, *where*? If not, how do you tell?

The second meteorite is coming in from the *lower left* along one branch of the hyperbola

4x

2

- y

2

- 80x = -340

Will it strike the Earth's surface? If so, *where (ordered pair)*? If not, how do you tell?

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: focus: directrix: vertex: focus: 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:

16

.

*x*

2

4

) (

2

*x x*

1

)

3

2

18

.

*x*

3

*x*

2

2

*x*

2

3

*x*

4

1

19.

3

*x*

2

*x*

3

7

*x*

*x*

2