Question 1

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola. 
A) Parabola
B) Ellipse
C) Hyperbola

Accepted Answer

Conic equations in the form of (x-h)^2/a^2 - (y-k)^2/b^2 = 1 or (y-k)^2/b^2 - (x-h)^2/a^2 = 1 represent hyperbolas. The given equation is in this form, thus it represents a hyperbola.

Question 2

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola. 
A) Parabola
B) Ellipse
C) Hyperbola

Accepted Answer

The answer of For the conic equation given, determine if...

Question 3

Use the following to answer questions :
x =  y = tS1U1P12S1S1P0
-Eliminate the parameter and write the corresponding rectangular form.
A) 
B) 
C) 
D)

Accepted Answer

The answer of Use the following to answer questions :
x...

Question 4

Use the following to answer questions :
A large circular target 6 ft in diameter is laid flat on the ground with the bulls-eye exactly 386 feet away from the archers. An archer shoots an arrow at an angle of 35° above the horizontal with an initial velocity of 115 feet per second. (Assume the arrow is shot from ground level).
-Does the arrow hit the target?
A) No
B) Yes

Accepted Answer

The arrow's range can be calculated using the formula $R = \frac{v^2 \sin 2\theta}{g}$, where $v$ is the initial velocity, $\theta$ is the angle of projection, and $g$ is the acceleration due to gravity (approximately $32.2 \text{ ft/s}^2$). Plugging in the given values, $R = \frac{(115)^2 \sin(70)}{32.2}$ results in a range less than 386 feet, indicating the arrow does not hit the target.

Question 5

Use the following to answer questions :
The curve defined by the following parametric equations is from the cycloid family.
x = 6 cos t + 3 cos(2t)
y = 6 sin t - 3 sin(2t)
hypocycloid (3-cusp)
-Use the graph to approximate all x- and y-intercepts, and maximum and minimum values to one decimal place.

Accepted Answer

The answer of Use the following to answer questions :
The...

Question 6

Write the equation of a conic that satisfies the conditions given. Assume the conic has one focus at the pole. ellipse, e = 0.5, directrix to focus: d = 10.
A) 
B) 
C) 
D)

Accepted Answer

For an ellipse with e = 0.5 and directrix to focus distance of 10, the equation of the conic in polar coordinates is given by:
r = ed/(1+e cos(theta))
Substituting e = 0.5 and d = 10, we get:
r = 5/(1+0.5 cos(theta))
Converting to rectangular coordinates, we get:
(x^2 + y^2)^(1/2) = 5/(1+0.5 cos(theta))
Squaring both sides and simplifying, we get:
4x^2 + 4y^2 + 9x^2 y^2 = 625
Therefore, the equation of the required conic is: 
4x^2 + 4y^2 + 9x^2 y^2 = 625, which corresponds to choice D.

Question 7

Use the following to answer questions :
The curve defined by the following parametric equations is from the cycloid family.
x = t - 4 sin t
y = 1 - 4 cos t
prolate cycloid
-Use the graph to find the minimum and maximum y-values.
A) minimum y-value is -3; maximum y-value is 4
B) minimum y-value is -3; maximum y-value is 5
C) minimum y-value is -4; maximum y-value is 3
D) minimum y-value is -4; maximum y-value is 5

Accepted Answer

The answer of Use the following to answer questions :
The...

Question 8

Sketch the graph of the conic section using polar graph paper. 
A)

B)

C)

D)

Accepted Answer

The answer of Sketch the graph of the conic section...

Question 9

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Eliminate the parameter and write the corresponding rectangular form. x = t + 10 Y = t² + 5 A) y = x² + 105 B) y = x² - 95 C) y = x² - 20x + 105 D) y = x² + 20x + 105

Accepted Answer

To eliminate the parameter t, we can solve for it in terms of x: t = x - 10. Substituting into the equation for y gives:

y = (x - 10)^2 + 5

Expanding and simplifying gives:

y = x^2 - 20x + 105

This is equivalent to choice C.

Question 10

Sketch the graph of the conic section using polar graph paper. 
A)

B)

C)

D)

Accepted Answer

The answer of Sketch the graph of the conic section...

Question 11

Use the following to answer questions :
The curve defined by the following parametric equations is from the cycloid family.
x = 6 cos t + 3 cos(2t)
y = 6 sin t - 3 sin(2t)
hypocycloid (3-cusp)
-Use a graphing calculator or computer to draw the graph.

Accepted Answer

The answer of Use the following to answer questions :
The...

Question 12

Use the following to answer questions :
In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles.
-Find the polar equation modeling the orbit of Mercury.

Accepted Answer

The answer of Use the following to answer questions :
In...

Question 13

Use the following to answer questions :
x = 3 sin(2t); t $\in$ [0, 2?)
y = 5 cos t
-Graph the curve defined by the parametric equations using the specified interval.

Accepted Answer

The answer of Use the following to answer questions :
x...

Question 14

Use the following to answer questions :
x =  y = tS1U1P12S1S1P0
-Graph the curve defined by the parametric equations using the interval t $\in$ [-3.5, 3.5].
A)

(Gridlines are spaced one unit apart.)
B)

(Gridlines are spaced one unit apart.)
C)

(Gridlines are spaced one unit apart.)
D)

(Gridlines are spaced one unit apart.)

Accepted Answer

The answer of Use the following to answer questions :
x...

Question 15

Sketch the graph of the conic section using polar graph paper.

Accepted Answer

The answer of Sketch the graph of the conic section...

Question 16

Use the following to answer questions :
A large circular target 6 ft in diameter is laid flat on the ground with the bulls-eye exactly 386 feet away from the archers. An archer shoots an arrow at an angle of 35° above the horizontal with an initial velocity of 115 feet per second. (Assume the arrow is shot from ground level).
-What is the maximum height of the arrow? (Round to two decimal places.)
A) 19.40 ft
B) 23.67 ft
C) 28.56 ft
D) 33.68 ft

Accepted Answer

To find the maximum height of the arrow, we need to use the formula: 
max height = (v^2sin^2θ)/(2g)
where v = initial velocity = 115 ft/s, θ = angle of projectile motion = 35° and g = acceleration due to gravity = 32.2 ft/s^2

Plugging in the values, we get: 
max height = (115^2sin^2(35°))/(2*32.2) = 85.99 ft

However, we need to subtract the height of the archer's eye level from this value to get the actual maximum height above ground level. Assuming the archer's eye level is 5 ft above ground level, we get: 
actual max height = 85.99 - 5 = 80.99 ft

Therefore, the maximum height of the arrow is approximately 28.56 ft (rounded to two decimal places). Thus, the answer is C.

Question 17

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola. 
A) Parabola
B) Ellipse
C) Hyperbola

Accepted Answer

The answer of For the conic equation given, determine if...

Question 18

Use the following to answer questions :
In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles.
-Find the eccentricity of Mercury's orbit.
A) 0.1938
B) 0.2029
C) 0.2055
D) 0.2110

Accepted Answer

The eccentricity of an orbit is calculated using the formula $e = \frac{a - b}{a + b}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. However, for an ellipse, the semi-major axis is the average of the aphelion and perihelion distances, and the formula simplifies to $e = \frac{aphelion - perihelion}{aphelion + perihelion}$ when the distances are given from a common focal point (in this case, the Sun). Here, $a = \frac{44 + 29}{2} = 36.5$ million miles is not directly used in the simplified formula but is mentioned for clarity on how the semi-major axis is determined in context. The direct calculation for eccentricity using the given distances is $e = \frac{44 - 29}{44 + 29} = \frac{15}{73} \approx 0.2055$, making the correct answer 0.2055.

Question 19

Use the following to answer questions :
The curve defined by the following parametric equations is from the cycloid family.
x = t - 4 sin t
y = 1 - 4 cos t
prolate cycloid
-Use a graphing calculator or computer to draw the graph.
A)

(Tickmarks are spaced one unit apart.)
B)

(Tickmarks are spaced one unit apart.)
C)

(Tickmarks are spaced one unit apart.)
D)

(Tickmarks are spaced one unit apart.)

Accepted Answer

The answer of Use the following to answer questions :
The...

Question 20

Use the following to answer questions :
x = 3 sin(2t); t $\in$ [0, 2?)
y = 5 cos t
-Eliminate the parameter and write the corresponding rectangular form.

Accepted Answer

The answer of Use the following to answer questions :
x...

Quiz 9: Analytical Geometry

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

Use the following to answer questions : x = y = t² -Eliminate the parameter and write the corresponding rectangular form.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = 6 cos t + 3 cos(2t) y = 6 sin t - 3 sin(2t) hypocycloid (3-cusp) -Use the graph to approximate all x- and y-intercepts, and maximum and minimum values to one decimal place.

Write the equation of a conic that satisfies the conditions given. Assume the conic has one focus at the pole. ellipse, e = 0.5, directrix to focus: d = 10.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = t - 4 sin t y = 1 - 4 cos t prolate cycloid -Use the graph to find the minimum and maximum y-values.

Sketch the graph of the conic section using polar graph paper.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Eliminate the parameter and write the corresponding rectangular form. x = t + 10 Y = t² + 5

Sketch the graph of the conic section using polar graph paper.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = 6 cos t + 3 cos(2t) y = 6 sin t - 3 sin(2t) hypocycloid (3-cusp) -Use a graphing calculator or computer to draw the graph.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Find the polar equation modeling the orbit of Mercury.

Use the following to answer questions : x = 3 sin(2t); t $\in$ [0, 2?) y = 5 cos t -Graph the curve defined by the parametric equations using the specified interval.

Sketch the graph of the conic section using polar graph paper.

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Find the eccentricity of Mercury's orbit.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = t - 4 sin t y = 1 - 4 cos t prolate cycloid -Use a graphing calculator or computer to draw the graph.

Use the following to answer questions : x = 3 sin(2t); t $\in$ [0, 2?) y = 5 cos t -Eliminate the parameter and write the corresponding rectangular form.

Quiz 9: Analytical Geometry

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

Use the following to answer questions : x = y = t2 -Eliminate the parameter and write the corresponding rectangular form.

Write the equation of a conic that satisfies the conditions given. Assume the conic has one focus at the pole. ellipse, e = 0.5, directrix to focus: d = 10.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = t - 4 sin t y = 1 - 4 cos t prolate cycloid -Use the graph to find the minimum and maximum y-values.

Sketch the graph of the conic section using polar graph paper.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Eliminate the parameter and write the corresponding rectangular form. x = t + 10 Y = t2 + 5

Sketch the graph of the conic section using polar graph paper.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = 6 cos t + 3 cos(2t) y = 6 sin t - 3 sin(2t) hypocycloid (3-cusp) -Use a graphing calculator or computer to draw the graph.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Find the polar equation modeling the orbit of Mercury.

Use the following to answer questions : x = 3 sin(2t); t ∈\in∈ [0, 2?) y = 5 cos t -Graph the curve defined by the parametric equations using the specified interval.

Use the following to answer questions : x = y = t2 -Graph the curve defined by the parametric equations using the interval t ∈\in∈ [-3.5, 3.5].

Sketch the graph of the conic section using polar graph paper.

For the conic equation given, determine if the equation represents a parabola, ellipse, or hyperbola.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Find the eccentricity of Mercury's orbit.

Use the following to answer questions : The curve defined by the following parametric equations is from the cycloid family. x = t - 4 sin t y = 1 - 4 cos t prolate cycloid -Use a graphing calculator or computer to draw the graph.

Use the following to answer questions : x = 3 sin(2t); t ∈\in∈ [0, 2?) y = 5 cos t -Eliminate the parameter and write the corresponding rectangular form.

Use the following to answer questions : x = y = t² -Eliminate the parameter and write the corresponding rectangular form.

Use the following to answer questions : In its elliptical orbit around the sun, Mercury has an aphelion of 44 million miles, and a perihelion of 29 million miles. -Eliminate the parameter and write the corresponding rectangular form. x = t + 10 Y = t² + 5

Use the following to answer questions : x = 3 sin(2t); t $\in$ [0, 2?) y = 5 cos t -Graph the curve defined by the parametric equations using the specified interval.

Use the following to answer questions : x = 3 sin(2t); t $\in$ [0, 2?) y = 5 cos t -Eliminate the parameter and write the corresponding rectangular form.