Question 1

Discuss the symmetries exhibited by an ellipse centered at the origin.

Accepted Answer

The answer of Discuss the symmetries exhibited by an ellipse...

Question 2

Discuss the graph of the equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b$

Accepted Answer

The answer of Discuss the graph of the equation \(\frac...

Question 3

An elliptical riding path is to be built on a rectangular piece of property that measures 10 mi by 6 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides 
A) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 100 } = 1$
B) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 6 } = 1$
D) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$

Accepted Answer

The answer of An elliptical riding path is to be...

Question 4

Graph the equation. Give the domain and range. Identify whether the equation is a function.
-$\frac{y}{6}=\sqrt{1-\frac{x^{2}}{16}}$
 
A)
$[ - 4,4 ] ; [ - 6,0 ]$; function

B)

$[ - 4,4 ] ; [ - 1,0 ] ;$ not a function
C)

$[ - 4,4 ] ; [ 0,6 ] ;$ function

D)

$[ - 4,4 ] ; [ 0,1 ]$; not a function

Accepted Answer

The answer of Graph the equation. Give the domain and...

Question 5

A rectangular board is 13 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.
A) 13 units
B) 26 units
C) 20 units
D) 40 units

Accepted Answer

The answer of A rectangular board is 13 by 20...

Question 6

Provide an appropriate response.
-If an ellipse has vertices at (-6, 0), (6, 0), (0, 2), and (0, -2), what is its domain? What is its range? 
A) Domain: $[ - 6,2 ]$; Range: $[ - 2,6 ]$
B) Domain: $[ - 6,6 ]$; Range: $[ - 2,2 ]$
C) Domain: $[ - 2,2 ]$; Range: $[ - 6,6 ]$
D) Domain: $[ 0,6 ]$; Range: $[ 0,2 ]$

Accepted Answer

The domain of an ellipse is determined by its vertices along the x-axis, which are (-6, 0) and (6, 0), so the domain is $[-6, 6]$. The range is determined by its vertices along the y-axis, which are (0, 2) and (0, -2), so the range is $[-2, 2]$.

Question 7

Provide an appropriate response.
-In the ellipse $\frac { ( y + 3 ) ^ { 2 } } { 36 } + \frac { ( x - 6 ) ^ { 2 } } { 16 } = 1$, what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?
A) Horizontal: $y = 3$; Vertical: $x = 6$
B) Horizontal: $y = 6$; Vertical: $x = - 3$
C) Horizontal: $y = 3$; Vertical: $x = - 6$
D) Horizontal: $y = - 3$; Vertical: $x = 6$

Accepted Answer

The horizontal axis of symmetry is at $y = -3$ and the vertical axis of symmetry is at $x = 6$, as these values correspond to the shifts in the ellipse equation.

Question 8

A railroad tunnel is shaped like a semi-ellipse. The height of the tunnel at the center is 25 ft and the vertical clearance must be 20 ft at a point 18 ft from the center. Find an equation for the ellipse. 
A) $\frac { x ^ { 2 } } { 324 } + \frac { y ^ { 2 } } { 625 } = 1$
B) $\frac { x ^ { 2 } } { 900 } + \frac { y ^ { 2 } } { 625 } = 1$
C) $\frac { x ^ { 2 } } { 900 } + \frac { y ^ { 2 } } { 400 } = 1$
D) $\frac { x ^ { 2 } } { 625 } + \frac { y ^ { 2 } } { 900 } = 1$

Accepted Answer

The answer of A railroad tunnel is shaped like a...

Question 9

Determine the two equations necessary to graph the ellipse with a graphing calculator.
-$\frac { x ^ { 2 } } { 25 } + \frac { ( y - 5 ) ^ { 2 } } { 49 } = 1$
A) $y = \frac { 7 \sqrt { 1225 - 25 x ^ { 2 } } } { 25 }$ and $y = - \frac { 7 \sqrt { 1225 - 25 x ^ { 2 } } } { 25 }$
B) $y = \frac { 25 - 7 \sqrt { 25 - x ^ { 2 } } } { 5 }$ and $y = \frac { 7 \sqrt { 25 - x ^ { 2 } } + 25 } { 5 }$
C) $y = \frac { \sqrt { 625 - 49 x ^ { 2 } } } { 5 }$ and $y = - \frac { \sqrt { 625 - 49 x ^ { 2 } } } { 5 }$
D) $y = \frac { \sqrt { 1850 - 49 x ^ { 2 } } } { 5 }$ and $y = - \frac { \sqrt { 1850 - 49 x ^ { 2 } } } { 5 }$

Accepted Answer

The answer of Determine the two equations necessary to graph...

Question 10

Find the eccentricity $ e $ of the ellipse.
-$$4 x ^ { 2 } + y ^ { 2 } = 36$$
A) $\sqrt { 3 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) $\frac { \sqrt { 5 } } { 2 }$
D) $\sqrt { 5 }$

Accepted Answer

The given equation can be rewritten as $\frac{x^2}{9} + \frac{y^2}{36} = 1$, which is in the standard form of an ellipse $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ with $a^2 = 36$ and $b^2 = 9$. The eccentricity $e$ is given by $e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{9}{36}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$.

Question 11

Find the eccentricity $ e $ of the ellipse.
-$$x ^ { 2 } + 4 y ^ { 2 } = 36$$
A) $\sqrt { 5 }$
B) $\frac { \sqrt { 5 } } { 2 }$
C) $\frac { \sqrt { 3 } } { 2 }$
D) $\sqrt { 3 }$

Accepted Answer

The answer of Find the eccentricity $ e $ of...

Question 12

Determine the two equations necessary to graph the ellipse with a graphing calculator.
-$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$
A) $y = \sqrt { \frac { 100 - x ^ { 2 } } { 5 } }$ and $y = - \sqrt { \frac { 80 - x ^ { 2 } } { 5 } }$
B) $y = \frac { 4 \sqrt { 25 - x ^ { 2 } } } { 25 }$ and $y = - \frac { 4 \sqrt { 25 - x ^ { 2 } } } { 25 }$
C) $y = \sqrt { \frac { 80 - x ^ { 2 } } { 4 } }$ and $y = - \sqrt { \frac { 80 - x ^ { 2 } } { 4 } }$
D) $y = \frac { 4 \sqrt { 25 - x ^ { 2 } } } { 5 }$ and $y = - \frac { 4 \sqrt { 25 - x ^ { 2 } } } { 5 }$

Accepted Answer

To graph the ellipse, solve for $y$ in terms of $x$. Starting with $\frac{x^2}{25} + \frac{y^2}{16} = 1$, multiply both sides by 16 to get $y^2 = 16(1 - \frac{x^2}{25})$. Taking the square root of both sides gives $y = \pm 4\sqrt{1 - \frac{x^2}{25}}$, which simplifies to $y = \pm \frac{4\sqrt{25 - x^2}}{5}$, matching choice D.

Question 13

Find the eccentricity $ e $ of the ellipse.
-$$x ^ { 2 } + 2 y ^ { 2 } = 6$$
A) $\sqrt { 2 }$
B) $\sqrt { 6 }$
C) $\frac { \sqrt { 6 } } { 2 }$
D) $\frac { \sqrt { 2 } } { 2 }$

Accepted Answer

The equation of the ellipse is given in the form $Ax^2 + By^2 = C$, where $A$ and $B$ are the coefficients of $x^2$ and $y^2$, respectively, and $C$ is a constant. To find the eccentricity $e$ of the ellipse, we first need to bring the equation to its standard form, which is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a^2$ and $b^2$ are the denominators of the $x^2$ and $y^2$ terms, respectively, representing the squares of the semi-major and semi-minor axes. For the given equation $x^2 + 2y^2 = 6$, dividing through by 6 gives $\frac{x^2}{6} + \frac{y^2}{3} = 1$, which means $a^2 = 6$ and $b^2 = 3$. The eccentricity $e$ of an ellipse is given by $e = \sqrt{1 - \frac{b^2}{a^2}}$.Substituting $a^2 = 6$ and $b^2 = 3$ into the formula gives $e = \sqrt{1 - \frac{3}{6}} = \sqrt{1 - \frac{1}{2}} = \sqrt{\frac{1}{2}} = \frac{\sqrt{2}}{2}$.

Question 14

Determine the two equations necessary to graph the ellipse with a graphing calculator.
-$$\frac { ( x - 5 ) ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 1$$
A) $y = \frac { 7 \sqrt { 36 - x ^ { 2 } } } { 6 }$ and $y = - \frac { 7 \sqrt { 36 - x ^ { 2 } } } { 6 }$
B) $y = \frac { 7 \sqrt { 900 - x ^ { 2 } } } { 6 }$ and $y = - \frac { 7 \sqrt { 900 - x ^ { 2 } } } { 6 }$
C) $y = \frac { 7 \sqrt { 25 - x ^ { 2 } } } { 6 }$ and $y = - \frac { 7 \sqrt { 25 - x ^ { 2 } } } { 6 }$
D) $y = \frac { 7 \sqrt { - x ^ { 2 } + 10 x + 11 } } { 6 }$ and $y = - \frac { 7 \sqrt { - x ^ { 2 } + 10 x + 11 } } { 6 }$

Accepted Answer

To graph the ellipse, we need to solve the given equation for $y$. The correct transformation involves completing the square for the $x$ term inside the square root, which matches option D. The original equation is $\frac{(x-5)^2}{36} + \frac{y^2}{49} = 1$. Solving for $y$ gives $y = \pm\sqrt{49 - \frac{49(x-5)^2}{36}}$, which simplifies to $y = \pm\frac{7\sqrt{36-(x-5)^2}}{6}$, matching the structure of option D after expanding $(x-5)^2$.

Question 15

Determine the two equations necessary to graph the ellipse with a graphing calculator.
-$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$$
A) $x = \sqrt { \frac { 192 - x ^ { 2 } } { 8 } }$ and $x = - \sqrt { \frac { 192 - y ^ { 2 } } { 8 } }$
B) $x = \frac { 3 \sqrt { 9 - y ^ { 2 } } } { 64 }$ and $x = - \frac { 3 \sqrt { 9 - y ^ { 2 } } } { 64 }$
C) $y = \frac { 8 \sqrt { 9 - x ^ { 2 } } } { 3 }$ and $y = - \frac { 8 \sqrt { 9 - x ^ { 2 } } } { 3 }$
D) $y = \frac { 8 \sqrt { 9 - x ^ { 2 } } } { 9 }$ and $y = - \frac { 8 \sqrt { 9 + x ^ { 2 } } } { 9 }$

Accepted Answer

To graph the ellipse $\frac{x^2}{9} + \frac{y^2}{64} = 1$ on a calculator, we need to solve for $y$ in terms of $x$. Isolating $y^2$, we get $y^2 = 64(1 - \frac{x^2}{9})$. Taking the square root of both sides gives $y = \pm \sqrt{64(1 - \frac{x^2}{9})}$, which simplifies to $y = \pm \frac{8\sqrt{9 - x^2}}{3}$, corresponding to the two equations in choice C.

Question 16

A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius of 578 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 490
Km to 261 km. 
A) $\frac { x ^ { 2 } } { 839 ^ { 2 } } + \frac { y ^ { 2 } } { 1068 ^ { 2 } } = 1$
B) $\frac { x ^ { 2 } } { 261 } + \frac { y ^ { 2 } } { 490 } = 1$
C) $\frac { x ^ { 2 } } { 1068 ^ { 2 } } + \frac { y ^ { 2 } } { 839 ^ { 2 } } = 1$
D) $\frac { x ^ { 2 } } { 490 } + \frac { y ^ { 2 } } { 261 } = 1$

Accepted Answer

The answer of A satellite is to be put into...

Question 17

Graph the equation. Give the domain and range. Identify whether the equation is a function.
-$y=-\sqrt{1-\frac{x^{2}}{36}}$

A)

$[-6,6] ;[-6,0] \text {; function }$
B)

$[ - 6,6 ] ; [ 0,1 ] ;$ not a function
C)

$[ - 6,6 ] ; [ 0,6 ] ;$ not a function

D)

$[ - 6,6 ] ; [ - 1,0 ]$; function

Accepted Answer

The answer of Graph the equation. Give the domain and...

Question 18

A rectangular board is 2 feet by 12 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.
A) 1.0 ft
B) 5.9 ft
C) 0.1 ft
D) 5.0 ft

Accepted Answer

The foci of an ellipse are located along the major axis, which in this case is the longer side of the rectangle, 12 feet. The distance from the center to each focus (c) can be found using the formula c = sqrt(a^2 - b^2), where 2a is the length of the major axis and 2b is the length of the minor axis. Here, a = 12/2 = 6 feet and b = 2/2 = 1 foot. Thus, c = sqrt(6^2 - 1^2) = sqrt(36 - 1) = sqrt(35) ≈ 5.9 feet.

Question 19

Find the eccentricity $ e $ of the ellipse.
-$$x ^ { 2 } + 25 y ^ { 2 } = 25$$
A) $2 \sqrt { 6 }$
B) $\frac { 2 \sqrt { 6 } } { 5 }$
C) $\frac { \sqrt { 26 } } { 5 }$
D) $\sqrt { 26 }$

Accepted Answer

The equation of the ellipse is in the standard form $\frac { x ^ { 2 } } { b ^ { 2 } } + \frac { y ^ { 2 } } { a ^ { 2 } } = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively. In this case, $a = 5$ and $b = 1$. The eccentricity is given by $e = \sqrt { 1 - \frac { b ^ { 2 } } { a ^ { 2 } } } = \sqrt { 1 - \frac { 1 } { 25 } } = \frac { 2 \sqrt { 6 } } { 5 }$.

Question 20

A rectangular board is 4 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?
A) 5.7 units
B) 2.0 units
C) 0.3 units
D) 4.0 units

Accepted Answer

The answer of A rectangular board is 4 units by...

Quiz 7: Arithmetic Sequence: Common Difference and First n Terms

Discuss the symmetries exhibited by an ellipse centered at the origin.

Discuss the graph of the equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b$

A rectangular board is 13 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.

Provide an appropriate response. -If an ellipse has vertices at (-6, 0) , (6, 0) , (0, 2) , and (0, -2) , what is its domain? What is its range?

Provide an appropriate response. -In the ellipse $\frac { ( y + 3 ) ^ { 2 } } { 36 } + \frac { ( x - 6 ) ^ { 2 } } { 16 } = 1$ , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 25 } + \frac { ( y - 5 ) ^ { 2 } } { 49 } = 1$

Find the eccentricity $e$ of the ellipse. - $4 x ^ { 2 } + y ^ { 2 } = 36$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 4 y ^ { 2 } = 36$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 2 y ^ { 2 } = 6$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { ( x - 5 ) ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 1$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$

A rectangular board is 2 feet by 12 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 25 y ^ { 2 } = 25$

A rectangular board is 4 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?

Quiz 7: Arithmetic Sequence: Common Difference and First n Terms

Discuss the symmetries exhibited by an ellipse centered at the origin.

Discuss the graph of the equation x2a2+y2b2=1 if a=b\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = ba2x2​+b2y2​=1 if a=b

An elliptical riding path is to be built on a rectangular piece of property that measures 10 mi by 6 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides

Graph the equation. Give the domain and range. Identify whether the equation is a function. - y6=1−x216\frac{y}{6}=\sqrt{1-\frac{x^{2}}{16}}6y​=1−16x2​​

A rectangular board is 13 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the length of the string.

Provide an appropriate response. -If an ellipse has vertices at (-6, 0) , (6, 0) , (0, 2) , and (0, -2) , what is its domain? What is its range?

Provide an appropriate response. -In the ellipse (y+3) 236+(x−6) 216=1\frac { ( y + 3 ) ^ { 2 } } { 36 } + \frac { ( x - 6 ) ^ { 2 } } { 16 } = 136(y+3) 2​+16(x−6) 2​=1 , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

A railroad tunnel is shaped like a semi-ellipse. The height of the tunnel at the center is 25 ft and the vertical clearance must be 20 ft at a point 18 ft from the center. Find an equation for the ellipse.

Determine the two equations necessary to graph the ellipse with a graphing calculator. - x225+(y−5) 249=1\frac { x ^ { 2 } } { 25 } + \frac { ( y - 5 ) ^ { 2 } } { 49 } = 125x2​+49(y−5) 2​=1

Find the eccentricity e e e of the ellipse. - 4x2+y2=364 x ^ { 2 } + y ^ { 2 } = 364x2+y2=36

Find the eccentricity e e e of the ellipse. - x2+4y2=36x ^ { 2 } + 4 y ^ { 2 } = 36x2+4y2=36

Determine the two equations necessary to graph the ellipse with a graphing calculator. - x225+y216=1\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 125x2​+16y2​=1

Find the eccentricity e e e of the ellipse. - x2+2y2=6x ^ { 2 } + 2 y ^ { 2 } = 6x2+2y2=6

Determine the two equations necessary to graph the ellipse with a graphing calculator. - (x−5) 236+y249=1\frac { ( x - 5 ) ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 136(x−5) 2​+49y2​=1

Determine the two equations necessary to graph the ellipse with a graphing calculator. - x29+y264=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 19x2​+64y2​=1

A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius of 578 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 490 Km to 261 km.

Graph the equation. Give the domain and range. Identify whether the equation is a function. - y=−1−x236y=-\sqrt{1-\frac{x^{2}}{36}}y=−1−36x2​​

A rectangular board is 2 feet by 12 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.

Find the eccentricity e e e of the ellipse. - x2+25y2=25x ^ { 2 } + 25 y ^ { 2 } = 25x2+25y2=25

A rectangular board is 4 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?

Discuss the graph of the equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b$

Provide an appropriate response. -In the ellipse $\frac { ( y + 3 ) ^ { 2 } } { 36 } + \frac { ( x - 6 ) ^ { 2 } } { 16 } = 1$ , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 25 } + \frac { ( y - 5 ) ^ { 2 } } { 49 } = 1$

Find the eccentricity $e$ of the ellipse. - $4 x ^ { 2 } + y ^ { 2 } = 36$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 4 y ^ { 2 } = 36$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 2 y ^ { 2 } = 6$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { ( x - 5 ) ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 49 } = 1$

Determine the two equations necessary to graph the ellipse with a graphing calculator. - $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 25 y ^ { 2 } = 25$