Question 1

an equation of the hyperbola with vertices

Accepted Answer

The equation of a hyperbola centered at the origin with horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $2a$ is the distance between the vertices, and the slopes of the asymptotes are $\pm \frac{b}{a}$. Given vertices at $(-7,0)$ and $(7,0)$, the distance between them is $14$, so $a = 7$. The slopes of the asymptotes are $\pm 2$, which means $\frac{b}{a} = 2$, so $b = 2a = 14$. Therefore, $a^2 = 49$ and $b^2 = 196$, leading to the equation $\frac{x^2}{49} - \frac{y^2}{196} = 1$.

Question 2

the center, foci, vertices, and eccentricity of the ellipse.

Accepted Answer

The center of the ellipse is given by the point $(h, k)$, which is $(3, -2)$ in this case. The lengths of the semi-major axis $a$ and semi-minor axis $b$ are given by the square roots of the denominators, so $a = 5$ and $b = 3$. The vertices are along the major axis, which is horizontal in this case, so they are $(-2, -2)$ and $(8, -2)$, found by adding and subtracting $a$ from the x-coordinate of the center. The foci are found by calculating $c = \sqrt{a^2 - b^2} = \sqrt{25 - 9} = \sqrt{16} = 4$, and since the major axis is horizontal, the foci are at $(3 - 4, -2)$ and $(3 + 4, -2)$, or $(-1, -2)$ and $(7, -2)$. The eccentricity $e$ is given by $e = c/a = 4/5$.

Question 3

an equation of the ellipse with vertices

Accepted Answer

Since the given ellipse has vertices at $( 0,8 ) , ( 14,8 )$, the major axis is parallel to the x-axis and has a length of 14 units. Thus, the distance between the center of the ellipse and one of the vertices is $\frac{1}{2} (14) = 7$ units. Also given is the eccentricity, $e = \frac{2}{7}$. Since $e < 1$, this tells us that the center of the ellipse is at the midpoint of the major axis. Thus, the center is at $(7,8)$. 
The distance between the center and each vertex is $7$ units. We can find the length of the minor axis using the equation $c^2 = a^2 - b^2$, where $a$ and $b$ are half the lengths of the major and minor axes, respectively, and $c$ is the distance between the center and each focus. Plugging in the known values, we have $7^2 = 7^2 - b^2$, which gives $b^2 = 7^2(1 - e^2) = 45$. Thus, $b = \sqrt{45} = 3\sqrt{5}$.
The equation of an ellipse centered at $(h,k)$ with vertical major axis of length $2a$ and horizontal minor axis of length $2b$ is given by $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$. Plugging in the known values, we get 
$$\frac{(x-7)^2}{(3\sqrt{5})^2} + \frac{(y-8)^2}{(7)^2} = 1$$
Simplifying, we get $\boxed{\frac{(x-7)^2}{49} + \frac{(y-8)^2}{45} = 1}$.

Question 4

the focus of the parabola given by

Accepted Answer

The given equation is in the form $(x - h) + (y - k)^2 = 0$, which is a parabola with its vertex at $(h, k)$ and opening leftward. The correct form for a parabola opening leftward is $(x - h) = -4p(y - k)^2$, where $p$ is the distance from the vertex to the focus. Comparing, we find $h = -4$, $k = 2$, and $p = \frac{1}{4}$. The focus of a leftward-opening parabola is at $(h - p, k)$, which gives $(-4 - \frac{1}{4}, 2) = \left(-\frac{17}{4}, 2\right)$.

Question 5

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Accepted Answer

The vertex form of a parabola that opens up or down is $x^2 = 4p(y - k)$, where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus or directrix. Here, the vertex is $(0, 8)$ and the directrix is $y = -3$, so the parabola opens upwards, and the distance $p$ is $8 - (-3) = 11$. Thus, the equation is $x^2 = 4(11)(y - 8)$ or $x^2 = 44(y - 8)$.

Question 6

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

Accepted Answer

The equation $5x^2 - 7y^2 + 2x + 2y - 7 = 0$ represents a hyperbola because it has two squared terms with opposite signs, which is characteristic of a hyperbola.

Question 7

Match the equation with its graph.

Accepted Answer

The answer of Match the equation with its graph. ...

Question 8

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

Accepted Answer

The given equation can be rearranged into the standard form of the equation of an ellipse: $\frac{(x+\frac{7}{14})^2}{\frac{1}{14^2}}+\frac{(y+\frac{2}{5})^2}{\frac{1}{10^2}}=1$. Therefore, the graph is an ellipse.

Question 9

the center of the ellipse given by

Accepted Answer

The center of an ellipse in the form $Ax^2 + By^2 = C$ is at the origin $(0,0)$ when not shifted by any terms involving $x$ or $y$. The given equation fits this form without any shifts.

Question 10

the vertices of the ellipse given by

Accepted Answer

To find the vertices, first rewrite the equation in standard form by completing the square. The given equation is $16(x^2 + 10x) + 9(y^2 - 2y) = -265$. Completing the square gives $(x+5)^2$ and $(y-1)^2$. The equation becomes $\frac{(x+5)^2}{9} + \frac{(y-1)^2}{16} = 1$, which is an ellipse centered at $(-5, 1)$ with semi-major axis 4 along the y-axis and semi-minor axis 3 along the x-axis. The vertices are $(-5, 1+4)$ and $(-5, 1-4)$, which are $(-5, 5)$ and $(-5, -3)$.

Question 11

Match the equation with its graph.

Accepted Answer

The answer of Match the equation with its graph. ...

Question 12

the vertex, focus, and directrix of the parabola and sketch its graph.

Accepted Answer

The answer of the vertex, focus, and directrix of the...

Question 13

Match the equation with its graph.

Accepted Answer

The answer of Match the equation with its graph. ...

Question 14

the foci of the ellipse given by

Accepted Answer

The answer of the foci of the ellipse given by...

Question 15

the directrix of the parabola given by

Accepted Answer

The answer of the directrix of the parabola given by...

Question 16

the vertex of the parabola given by

Accepted Answer

The answer of the vertex of the parabola given by...

Question 17

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Accepted Answer

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Question 18

Match the equation with its graph.

Accepted Answer

The answer of Match the equation with its graph. ...

Question 19

the eccentricity of the ellipse given by

Accepted Answer

The answer of the eccentricity of the ellipse given by...

Question 20

the center, foci, and vertices of the hyperbola.

Accepted Answer

The answer of the center, foci, and vertices of the...

Question 21

the result, &#34;the set of parametric equations for the ellipse is

Accepted Answer

The answer of the result, &#34;the set of parametric equations...

Question 22

the result, &#34;the set of parametric equations for the circle is

Accepted Answer

The answer of the result, &#34;the set of parametric equations...

Question 23

a set of parametric equations for the rectangular equation

Accepted Answer

The answer of a set of parametric equations for the...

Question 24

Sketch the curve represented by the parametric equations

Accepted Answer

The answer of Sketch the curve represented by the parametric...

Question 25

Suppose a cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 240 meters apart and 40 meters above the roadway as shown in the figure
Given below. The cable touches the roadway midway between the towers. Find the length of the
Parabolic supporting cable. Round your answer to two decimal places.

Accepted Answer

The answer of Suppose a cable of a suspension bridge...

Question 26

the result, &#34;the set of parametric equations for the line passing through

Accepted Answer

The answer of the result, &#34;the set of parametric equations...

Question 27

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

Accepted Answer

The answer of Classify the graph of the equation as...

Question 28

Write the corresponding rectangular equation for the curve represented by the parametric equations   by eliminating the parameter.

Accepted Answer

The answer of Write the corresponding rectangular equation for the...

Question 29

Write the corresponding rectangular equation for the curve represented by the parametric equation   by eliminating the parameter.

Accepted Answer

The answer of Write the corresponding rectangular equation for the...

Question 30

Suppose a cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 240 meters apart and 40 meters above the roadway as shown in the figure
Given below. The cable touches the roadway midway between the towers. Find an equation for the
Parabolic shape of cable.

Accepted Answer

The answer of Suppose a cable of a suspension bridge...

Question 31

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter.

Accepted Answer

The answer of Sketch the curve represented by the parametric...

Question 32

Earth moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the maximum
Distance (aphelion) of Earth from the sun. Round your answer to nearest kilometer.

A)146,606,040 km
B)2,498,287 km
C)152,589,960 km
D)152,096,287 km
E)147,099,713 km

Accepted Answer

The answer of Earth moves in an elliptical orbit with...

Question 33

a set of parametric equations for the rectangular equation

Accepted Answer

The answer of a set of parametric equations for the...

Question 34

Sketch the curve represented by the parametric equations

Accepted Answer

The answer of Sketch the curve represented by the parametric...

Question 35

a set of parametric equations for the rectangular equation

Accepted Answer

The answer of a set of parametric equations for the...

Question 36

Write the corresponding rectangular equation for the curve represented by the parametric equations

Accepted Answer

The answer of Write the corresponding rectangular equation for the...

Question 37

Uranus moves in an elliptical orbit with the sun at one of the foci. The length of the half of the major axis is 2,876,769,540 kilometers, and the eccentricity is 0.0444. Find the minimum
Distance (perihelion) of Uranus from the sun. Round your answer to nearest kilometer.

A)3,004,498,108 km
B)2,749,040,972 km
C)2,819,234,149 km
D)127,728,568 km
E)2,934,304,931 km

Accepted Answer

The answer of Uranus moves in an elliptical orbit with...

Question 38

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

Accepted Answer

The answer of Classify the graph of the equation as...

Question 39

Suppose a church window is bounded above by a parabola and below by the arc of a circle as shown in figure given below. Find the surface area of the window. Round your answer to
Three decimal places.

Accepted Answer

The answer of Suppose a church window is bounded above...

Question 40

Sketch the curve represented by the parametric equations, and write the corresponding rectangular equation by eliminating the parameter.

Accepted Answer

The answer of Sketch the curve represented by the parametric...

Question 41

the arc length of the curve on the given interval.

Accepted Answer

The answer of the arc length of the curve on...

Question 42

the arc length of the curve   your answer to three decimal places.&#10;A)287.453&#10;B)191.635&#10;C)193.606&#10;D)66.480&#10;E)99.721

Accepted Answer

The answer of the arc length of the curve ...

Question 43

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Accepted Answer

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Question 44

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Accepted Answer

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Question 45

the arc length of the curve on the given interval.

Accepted Answer

The answer of the arc length of the curve on...

Question 46

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Accepted Answer

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Question 47

Round your answer to two decimal places, if necessary.

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The answer of   Round your answer to two...

Question 48

the arc length of the curve on the given interval.

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The answer of the arc length of the curve on...

Question 49

point corresponding to t = 5.

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The answer of   point corresponding to t =...

Question 50

or concave upward.

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The answer of   or concave upward.  ...

Question 51

parametric equations for the path of a projectile launched at a height h feet above

Accepted Answer

The answer of parametric equations for the path of a...

Question 52

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Accepted Answer

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Question 53

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Accepted Answer

The answer of 11eb8bd6_fbda_4536_8494_ed9960f96ec5_TB8682_00 11eb8bd6_fbda_4537_8494_fd424d55be79_TB8682_00...

Question 54

Determine the t intervals on which the curve   downward or concave upward.

Accepted Answer

The answer of Determine the t intervals on which the...

Question 55

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Accepted Answer

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