Math 2412 Review Items for Exam 2
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You may turn in your work for this review at the time that you take the exam in order to earn 5 points extra credit.
Graph the ellipse and locate the foci.
1) 9x2 = 144 - 16y 2
y
10
5
-10
-5
5
10 x
-5
-10
Solve the problem.
2) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the
roadway is 36 feet and the height of the arch over the center of the roadway is 13 feet. Two trucks plan to
use this road. They are both 12 feet wide. Truck 1 has an overall height of 12 feet and Truck 2 has an overall
height of 13 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under
the bridge.
Use vertices and asymptotes to graph the hyperbola. Find the equations of the asymptotes.
2
2
3) y - x = 1
9
36
y
10
5
-10
-5
5
10 x
-5
-10
Solve the problem.
4) Two recording devices are set 2200 feet apart, with the device at point A to the west of the device at point B.
At a point on a line between the devices, 200 feet from point B, a small amount of explosive is detonated.
The recording devices record the time the sound reaches each one. How far directly north of site B should a
second explosion be done so that the measured time difference recorded by the devices is the same as that
for the first detonation?
1
Match the equation to the graph.
5) y 2 = 11x
A)
B)
y
y
5
5
-5
5
x
-5
-5
5
x
5
x
-5
C)
D)
y
y
5
-5
5
5
x
-5
-5
-5
Solve the problem.
6) An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower
down to the roadway, just touching it there, and up again to the top of a second tower. The towers are both
16 inches tall and stand 80 inches apart. At some point along the road from the lowest point of the cable, the
cable is 1.44 inches above the roadway. Find the distance between that point and the base of the nearest
tower.
2
7) A satellite dish is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the
dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of
16 feet and a depth of 5 feet. The parabola is positioned in a rectangular coordinate system with its vertex at
the origin. The receiver should be placed at the focus (0, p). The value of p is given by the equation a = 1 .
4p
How far from the base of the dish should the receiver be placed?
(8, 5)
5 feet
Identify the equation without completing the square.
8) 4x2 - 4x + y + 3 = 0
9) 3x2 + 4y 2 + 5x - 2y = 0
10) 4x2 - 2y 2 + 7x + 3y + 1 = 0
Write the equation in terms of a rotated x'y'-system using !, the angle of rotation. Write the equation involving x' and
y' in standard form.
11) 5x2 - 6xy + 5y 2 - 8 = 0; ! = 45°
Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term.
12) x2 + 2xy + y 2 - 8x + 8y = 0
Rewrite the equation in a rotated x'y'-system without an x'y' term. Express the equation involving x' and y' in the
standard form of a conic section.
13) x2 + 2xy + y 2 + x - y - 4 = 0
3
Use the rotated system to graph the equation.
14) 7x2 + 2xy + 7y 2 = 24
4
2
-4
-2
2
4
-2
-4
Identify the equation without applying a rotation of axes.
15) x2 - 4xy - 3y 2 - 2x + 4y - 3 = 0
16) 9x2 + 9xy + 4y 2 + 3x - 2y - 3 = 0
17) 6x2 - 9xy + 4y 2 - 3x + 6 = 0
Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve
described by the parametric equations corresponding to the given value of t.
18) x = t3 + 1, y = 6 - t4; t = 2
19) x = (40 cos 30°)t, y = 3 + (40 sin 30°)t - 13t2; t = 4
Use point plotting to graph the plane curve described by the given parametric equations.
20) x = t2, y = t + 3; 0 ≤ t ≤ 4
y
-20
20 x
Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.
21) x = 9 sin t, y = 9 cos t; 0 ≤ t ≤ 2π
4
22) x = t3 + 1, y = t3 - 15; -2 ≤ t ≤ 2
23) x = et, y = e-t; -∞ < t < ∞
Find a set of parametric equations for the conic section.
24) Circle: Center: (5, 2); Radius: 2
Find a set of parametric equations for the rectangular equation.
25) (x - 2)2 = 2y
Solve the problem.
26) A baseball pitcher throws a baseball with an initial velocity of 134 feet per second at an angle of 20° to the
horizontal. The ball leaves the pitcher's hand at a height of 5 feet. Find parametric equations that describe
the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum
height? What is the maximum height of the ball?
Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located
at the pole.
2
27) r =
2 + 2 sin !
28) r =
8
8 - 4 sin !
Graph the polar equation.
9
29) r =
3 - 3 cos !
Identify the directrix and vertex.
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1
2
3
4
5 r
5
30) r =
4
4 - cos !
Identify the directrix and vertices.
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1
2
3
4
5 r
Solve the problem.
31) Halley's comet has an elliptical orbit with the sun at one focus. Its orbit shown below is given
10.48
approximately by r =
. In the formula, r is measured in astronomical units. (One astronomical
1 + 0.838 sin !
unit is the average distance from Earth to the sun, approximately 93 million miles.)
Find the distance from Halley's comet to the sun at its shortest distance from the sun. Round to the nearest
hundredth of an astronomical unit and the nearest million miles.
6
Answer Key
Testname: 2412REVIEW2
1) foci at (
7, 0) and (-
14)
7, 0)
y
-10
10
4
5
2
-5
5
-4
10 x
-2
2
-5
-2
-10
-4
4
15) hyperbola
16) ellipse
17) parabola
18) (9, -10)
2) Truck 1 can pass under the bridge, but Truck 2
cannot.
3) Asymptotes: y = ± 1 x
2
19) (80
20)
y
10
3, -125)
y
5
-10
-5
5
10 x
-20
-5
20 x
-10
4) 444.44 feet
5) A
6) 28 in.
7) 3 1 feet from the base
5
21) x2 + y 2 = 81; -9 ≤ x ≤ 9
22) y = x - 16; -7 ≤ x ≤ 9
23) y = e-ln x; 0 < x < ∞
24) x = 5 + 2 cos t; y = 2 + 2 sin t
2
25) x = t + 2; y = t
2
8) parabola
9) ellipse
10) hyperbola
2
11) x' + y' 2 = 1
4
12) x =
13) x'2 =
2
2 (x' - y'); y =
2
2
26) x = 125.92t; y = -16t2 + 45.83t + 5;
2.97 sec;
1.432 sec;
37.819 feet
27) parabola; The directrix is 1 unit(s) above the pole at
y = 1.
28) ellipse; The directrix is 2 unit(s) below the pole at y
= - 2.
2 (x' + y')
2 y' + 2
7
Answer Key
Testname: 2412REVIEW2
29) directrix: 3 unit(s) to the left of
the pole at x = -3
vertex: 3 , π
2
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1
2
3
4
5 r
30) directrix: 4 unit(s) to the left of
the pole at x = -4
vertices: 4 , π , 4 , 0
5
3
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
1
2
3
4
5 r
31) 5.7 astronomical units; 530 million miles
8