Deck 9: Conics, Systems of Nonlinear Equations and Inequalities, and Parametric Equations

Question 1

Graph the curve defined by the parametric equations.
x= t , y= t² , t in [-2, 2]

Answer 1

Question 2

Graph the curve defined by the parametric equations.
, ,t in [0, 2π]

Answer 2

Question 3

Graph the curve defined by the parametric equations.
,y=t,t in [0, 2π]

Answer 3

Question 4

Graph the curve defined by the parametric equations.
, ,t in [0, 1]

Question 5

Graph the curve defined by the parametric equations.
, ,t in [0, 2π]

Question 6

Graph the curve defined by the parametric equations.
, ,t in [0, 2π]

Question 7

Graph the curve defined by the parametric equations.
, ,t in [-π/4,π/4]

Question 8

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

Question 9

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

Question 10

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

Question 11

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.

Question 12

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

Question 13

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 560 ft/sec at an angle of 47° with the horizontal. After how many seconds does the projectile hit the ground? Round to one decimal place.

Question 14

Recall that the flight of a projectile can be modeled with the parametric equations:
,
where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 465 ft/sec at an angle of 33° with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is the maximum altitude? Round to the nearest integer.

Question 15

Recall that the flight of a projectile can be modeled with the parametric equations:
,
where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A baseball is hit at an initial speed of 115 mph and angle of 32° at a height of 2 feet above the ground. If home plate is 425 feet from the back fence, which is 12 feet tall, will the baseball clear the back fence for a home run?

Question 16

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A baseball is hit at an initial speed of 110 mph and angle of 31° at a height of 2 feet above the ground. If there's no back fence or other obstruction, how far does the baseball travel (horizontal distance), and what is the maximum height? Round to one decimal place.

Question 17

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A gun is fired at an angle of 21°, and the bullet has an initial speed of 875 feet per second. How high does the bullet go, and when does it hit the ground? Round the altitude to the nearest foot and time to the nearest tenth of a second.

Question 18

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A missile is fired from a ship at an angle of 32°, an initial height of 20 feet above the water's surface, and at a speed of 4450 feet per second. How long will it be before the missile hits the water? Round to the nearest tenth of a second.

Question 19

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A missile is fired from a ship at an angle of 56°, an initial height of 18 feet above the water's surface, and at a speed of 4430 feet per second. Will it be able to hit a target that is 2.4 miles away?

Question 20

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, ν_o is the initial velocity, θ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 200 ft/sec at an angle of 30° with the horizontal. Plot the path of the projectile on a graph.

Question 21

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 22

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 23

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 24

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 25

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 26

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 27

Graph the curve defined by the parametric equations.

A)
B)
C)
D)

Question 28

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

A)
B)
C)
D)

Question 29

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

A)
B)
C)
D)

Question 30

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

A)
B)
C)
D)

Question 31

The given parametric equations define a plane curve. Find an equation in rectangular form that also corresponds to the plane curve.
,

A)
B)
C)
D)

Question 32

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 420 ft/sec at an angle of 37° with the horizontal. After how many seconds does the projectile hit the ground? Round to one decimal place.

A) 15.8 seconds
B) 31.6 seconds
C) 7.9 seconds
D) 21.0 seconds

Question 33

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 520 ft/sec at an angle of 31° with the horizontal. How far does the projectile travel (what is the horizontal distance), and what is the maximum altitude? Round to the nearest integer.

A) 7,461 feet; 1,121 feet
B) 14,922 feet; 1,121 feet
C) 7,461 feet; 2,241 feet
D) 14,922 feet; 2,241 feet

Question 34

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A baseball is hit at an initial speed of 100 mph and angle of 32° at a height of 3 feet above the ground. If home plate is 465 feet from the back fence, which is 19 feet tall, will the baseball clear the back fence for a home run?

A) Yes. The height will be over 19 feet at that time.
B) No. The height will not be over 19 feet at that time.
C) no solution

Question 35

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A baseball is hit at an initial speed of 110 mph and angle of 24° at a height of 5 feet above the ground. If there's no back fence or other obstruction, how far does the baseball travel (horizontal distance), and what is the maximum height? Round to one decimal place.

A) 615.5 feet; 72.3 feet
B) 604.5 feet; 72.3 feet
C) 615.5 feet; 67.3 feet
D) 604.5 feet; 165.4 feet

Question 36

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A gun is fired at an angle of 23°, and the bullet has an initial speed of 2345 feet per second. How high does the bullet go, and when does it hit the ground? Round the altitude to the nearest foot and time to the nearest tenth of a second.

A) 13,118 feet; 57.3 seconds
B) 26,236 feet; 57.3 seconds
C) 13,118 feet; 29 seconds
D) 33,573 feet; 29 seconds

Question 37

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A missile is fired from a ship at an angle of 31°, an initial height of 13 feet above the water's surface, and at a speed of 5500 feet per second. How long will it be before the missile hits the water? Round to one decimal place.

A) 177.0 seconds
B) 354.1 seconds
C) 88.5 seconds
D) 294.7 seconds

Question 38

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A missile is fired from a ship at an angle of 39°, an initial height of 20 feet above the water's surface, and at a speed of 3890 feet per second. Will it be able to hit a target that is 2.4 miles away?

A) Yes, it will easily hit the target.
B) No, it will miss the target.
C) no solution

Question 39

Recall that the flight of a projectile can be modeled with the parametric equations:
, where t is in seconds, v_o is the initial velocity, $\theta$ is the angle with the horizontal, and x and y are in feet.
A projectile is launched at a speed of 200 ft/sec at an angle of 60° with the horizontal. Plot the path of the projectile on a graph.

A)
B)
C)
D)

Question 40

The given rectangular equation defines a plane curve. Find the parametric equations that also corresponds to the plane curve.

Question 41

The given rectangular equation defines a plane curve. Find the parametric equations that also corresponds to the plane curve.

Question 42

The given rectangular equation defines a plane curve. Find the parametric equations that also corresponds to the plane curve.

Question 43

The given rectangular equation defines a plane curve. Find the parametric equations that also corresponds to the plane curve.

Question 44

The given rectangular equation defines a plane curve. Find the parametric equations that also corresponds to the plane curve.

Question 45

Find the polar equation that represents the conic described (assume the focus is at the origin).
Ellipse with eccentricity e =

and directrix

Question 46

Find the polar equation that represents the conic described (assume the focus is at the origin).
Hyperbola with eccentricity e =

and directrix

Question 47

Find the polar equation that represents the conic described (assume the focus is at the origin).
Parabola with eccentricity e =1 and directrix

Question 48

Identify the conic that the polar equation represents.

Question 49

Identify the conic that the polar equation represents.

Question 50

Identify the conic that the polar equation represents.

Question 51

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 52

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 53

For the polar equation, graph the conic.

A)
B)
C)
D)

Question 54

For the polar equation, graph the conic.

A)
B)
C)
D)

Question 55

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 56

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 57

For the polar equation, graph the conic.

A)
B)
C)
D)

Question 58

For the polar equation, graph the conic.

A)

B)

C)

D)

Question 59

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 60

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 61

For the polar equation, (a) Identify the conic as either a parabola, ellipse, or hyperbola; (b) find the eccentricity and vertex (or vertices); and (c) graph.

Question 62

For the polar equation, graph the conic.

A)
B)
C)
D)

Question 63

For the polar equation, graph the conic.

A)
B)
C)
D)

Question 64

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 65

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 66

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 67

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 68

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 69

Identify the conic (parabola, ellipse, or hyperbola) that is represented by the equation.

Question 70

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

Question 71

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

Question 72

If the xy-coordinate axes are rotated 45°, find the XY coordinates of the point (x, y) = (3, -6).

Question 73

If the xy-coordinate axes are rotated 30°, find the XY coordinates of the point (x, y) =

.

Question 74

If the xy-coordinate axes are rotated 60°, find the XY coordinates of the point (x, y) =

.

Question 75

If the xy-coordinate axes are rotated 45°, find the XY coordinates of the point (x, y) = (-33, 3).

Question 76

If the xy-coordinate axes are rotated 30°, find the XY coordinates of the point (x, y) =

.

Question 77

If the xy-coordinate axes are rotated 60°, find the XY coordinates of the point (x, y) =

.

Question 78

Determine the angle of rotation necessary to transform the equation in x and y into an equation in X and Y with no XY term. Round to one decimal place.

Question 79

(a) Identify the type of conic by the discriminant, (b) transform the equation in x and y into an equation in X and Y (without an XY term) by rotating the x- and y-axes by an angle of θ to arrive at the new X- and Y-axes, and (c) graph the resulting equation (showing both sets of axes).

Question 80

Graph the second-degree equation. (Hint: Transform the equation into an equation that contains no xy term).