Calculus Academic Essay

QUESTION 1
1.    Find the vertices and locate the foci for the hyperbola whose equation is given.

81y2 – 64×2 = 5184

vertices: (-8, 0), (8, 0)
foci: (- , 0), ( , 0)

vertices: (0, -8), (0, 8)
foci: (0, – ), (0,  )

vertices: (0, -9), (0, 9)
foci: (0, – ), (0,  )

vertices: (-9, 0), (9, 0)
foci: (- , 0), ( , 0)

4 points
QUESTION 2
1.    Find the vertices and locate the foci for the hyperbola whose equation is given.

49×2 – 16y2 = 784

vertices: (-4, 0), (4, 0)
foci: (- , 0), ( , 0)

vertices: (0, -4), (0, 4)
foci: (0, – ), (0,  )

vertices: (-4, 0), (4, 0)
foci: (- , 0), ( , 0)

vertices: (-7, 0), (7, 0)
foci: (- , 0), ( , 0)

4 points
QUESTION 3
1.    Find the standard form of the equation of the hyperbola satisfying the given conditions.

Center: (6, 5); Focus: (3, 5); Vertex: (5, 5)

– (y – 6)2 = 1

– (y – 5)2 = 1

(x – 5)2 –   = 1

(x – 6)2 –   = 1

4 points
QUESTION 4
1.    Solve the problem.

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 6.25 inches tall and stand 50 inches apart. At some point along the road from the lowest point of the cable, the cable is 1 inches above the roadway. Find the distance between that point and the base of the nearest tower.

10.2 in.

15 in.

9.8 in.

15.2 in.
4 points
QUESTION 5
1.    Solve the problem.

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 stand 40 inches apart. At a point between the towers and 10 inches along the road from the base of one tower, the cable is 1 inches above the roadway. Find the height of the towers.

4 in.

4.5 in.

6 in.

3.5 in.
4 points
QUESTION 6
1.    Find the standard form of the equation of the ellipse satisfying the given conditions.

Major axis vertical with length 16; length of minor axis = 6; center (0, 0)

+   = 1

+   = 1

+   = 1

+   = 1

4 points
QUESTION 7
1.    Identify the equation as a parabola, circle, ellipse, or hyperbola.

12y = 3(x + 8)2

Circle

Hyperbola

Parabola

Ellipse
4 points
QUESTION 8
1.    Find the vertices and locate the foci for the hyperbola whose equation is given.

y = ±

vertices: (0, -2 ), (0, 2 )
foci: (0, -2 ), (0, 2 )

vertices: (-2 , 0), (2 , 0)
foci: (-2 , 0), (2 , 0)

vertices: (-12, 0), (12, 0)
foci: (-2 , 0), (2 , 0)

vertices: (-12, 0), (12, 0)
foci: (-2 , 0), (2 , 0)

4 points
QUESTION 9
1.    Find the standard form of the equation of the ellipse satisfying the given conditions.

Foci: (0, -2), (0, 2); y-intercepts: -3 and 3

+   = 1

+   = 1

+   = 1

+   = 1

4 points
QUESTION 10
1.    Find the standard form of the equation of the parabola using the information given.

Vertex: (4, -7); Focus: (3, -7)

(y + 7)2 = -4(x – 4)

(x + 4)2 = -16(y – 7)

(x + 4)2 = 16(y – 7)

(y + 7)2 = 4(x – 4)
4 points
QUESTION 11
1.    Find the standard form of the equation of the hyperbola satisfying the given conditions.

Endpoints of transverse axis: (0, -10), (0, 10); asymptote: y =  x

–   = 1

–   = 1

–   = 1

–   = 1

4 points
QUESTION 12
1.    Convert the equation to the standard form for a hyperbola by completing the square on x and y.

4y2 – 25×2 – 16y + 100x – 184 = 0

–   = 1

–   = 1

–   = 1

–   = 1

4 points
QUESTION 13
1.    Find the standard form of the equation of the ellipse satisfying the given conditions.

Major axis horizontal with length 12; length of minor axis = 6; center (0, 0)

+   = 1

+   = 1

+   = 1

+   = 1

4 points
QUESTION 14
1.    Find the standard form of the equation of the parabola using the information given.

Focus: (3, 3); Directrix: y = -5

(x – 3)2 = 16(y + 1)

(y + 1)2 = 16(x – 3)

(y – 3)2 = 16(x + 1)

(x + 1)2 = 16(y – 3)
4 points
QUESTION 15
1.    Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate.

x2 – 6x – 6y – 21 = 0

(x + 3)2 = 6(y + 5)

(x – 3)2 = 6(y – 5)

(x – 3)2 = 6(y + 5)

(x + 3)2 = -6(y + 5)
4 points
QUESTION 16
1.    Convert the equation to the standard form for a hyperbola by completing the square on x and y.

4×2 – 25y2 – 8x + 50y – 121 = 0

–   = 1

–   = 1

–   = 1

–   = 1

4 points
QUESTION 17
1.    Find the standard form of the equation of the parabola using the information given.

Focus: (-3, -1); Directrix: x = 7

(x – 2)2 = -20(y + 1)

(y + 1)2 = -20(x – 2)

(y – 2)2 = -20(x + 1)

(x + 1)2 = -20(y – 2)
4 points
QUESTION 18
1.    Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate.

y2 – 4y – 2x – 2 = 0

(y – 2)2 = 2(x + 3)

(y + 2)2 = -2(x + 3)

(y + 2)2 = 2(x + 3)

(y – 2)2 = 2(x – 3)
4 points
QUESTION 19
1.    Identify the equation as a parabola, circle, ellipse, or hyperbola.

4×2 = 36 – 4y2

Parabola

Hyperbola

Ellipse

Circle
4 points
QUESTION 20
1.    Find the standard form of the equation of the ellipse satisfying the given conditions.

Endpoints of major axis: (10, -3) and (-2, -3); endpoints of minor axis: (4, -1) and (4, -5)

+   = 1

+   = 1

+   = 0

+   = 1

4 points
QUESTION 21
1.    Identify the equation as a parabola, circle, ellipse, or hyperbola.

2x = 2y2 – 30

Ellipse

Circle

Parabola

Hyperbola
4 points
QUESTION 22
1.    Find the standard form of the equation of the hyperbola satisfying the given conditions.

Endpoints of transverse axis: (-6, 0), (6, 0); foci: (-7, 0), (-7, 0)

–   = 1

–   = 1

–   = 1

–   = 1

4 points
QUESTION 23
1.    Find the standard form of the equation of the ellipse satisfying the given conditions.

Endpoints of major axis:   and  ; endpoints of minor axis:   and

+   = 1

+   = 1

+   = 1

+   = 1

4 points
QUESTION 24
1.    Identify the equation as a parabola, circle, ellipse, or hyperbola.

(x – 2)2 = 16 – y2

Circle

Ellipse

Hyperbola

Parabola
4 points
QUESTION 25
1.    Identify the equation as a parabola, circle, ellipse, or hyperbola.

9×2 = 4y2 + 36

Hyperbola

Ellipse

Parabola

Circle

 

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