Math Help

 

 

 

 

1.         Solve the triangle.

A = 18°, C = 104°, c = 5 (5 points)

 

           

            B = 32°, a ≈ 15.7, b ≈ 13.7

            B = 58°, a ≈ 15.7, b ≈ 13.7

            B = 58°, a ≈ 1.6, b ≈ 4.4

            B = 58°, a ≈ 15.7, b ≈ 4.4

 

2.         State whether the given measurements determine zero, one, or two triangles.

C = 37°, a = 16, c = 14 (5 points)

 

           

            Zero

            Two

            One

 

3.         Two triangles can be formed with the given information. Use the Law of Sines to solve the triangles.

B = 49°, a = 16, b = 14 (5 points)

 

           

            A = 30.4°, C = 100.6°, c = 18.2; A = 149.6°, C = 79.4°, c = 18.2

            A = 30.4°, C = 100.6°, c = 10.7; A = 149.6°, C = 79.4°, c = 10.7

            A = 59.6°, C = 71.4°, c = 11.1; A = 120.4°, C = 10.6°, c = 11.1

            A = 59.6°, C = 71.4°, c = 17.6; A = 120.4°, C = 10.6°, c = 3.4

 

4.         Solve the triangle.

B = 36°, a = 38, c = 17 (5 points)

 

           

            b ≈ 41.3, C ≈ 22.3, A ≈ 121.7

            b ≈ 41.3, C ≈ 26.3, A ≈ 117.7

            b ≈ 26.2, C ≈ 118.7, A ≈ 25.3

            b ≈ 26.2, C ≈ 22.3, A ≈ 121.7

5.         Find the area of the triangle with the given measurements. Round the solution to the nearest hundredth if necessary.

B = 74°, a = 14 cm, c = 20 cm (5 points)

 

           

            38.59 cm2

            134.58 cm2

            140 cm2

            269.15 cm2

 

6.         Find the area of the triangle with the given measurements. Round the solution to the nearest hundredth if necessary.

B = 67°, a = 10 cm, c = 20 cm (5 points)

 

           

            92.05 cm2

            184.1 cm2

            100 cm2

            39.07 cm2

 

7.         Given that P = (-5, 5) and Q = (-13, 10), find the component form and magnitude of 2 vector PQ. (5 points)

           

            <-16, 10>, square root of three hundred and fifty six

            <-36, 10>, 2 square root of 349

            <-16, 10>, square root of 178

            <16, -10>, 2 square root 89

 

8.         Let u = <-6, -2>, v = <-2, 3>. Find -3u + 2v. (5 points)

           

            <14, 12>

            <22, 0>

            <14, 1>

            <24, -3>

 

9.         Find a • b.

a = 5i + 7j, b = -4i + 3j (5 points)

 

           

            <1, 10>

            <-20, 21>

            1

            41

 

10.       Determine whether the vectors u and v are parallel, orthogonal, or neither.

u = <10, 0>, v = <0, -9> (5 points)

 

           

            Parallel

            Orthogonal

            Neither

 

11.       Find the first six terms of the sequence.

a1 = -7, an = 2 • an-1 (5 points)

 

           

            -7, -14, -12, -10, -8, -6

            0, 2, -14, -12, -10, -8

            -7, -14, -28, -56, -112, -224

            -14, -28, -56, -112, -224, -448

 

12.       Determine whether the sequence converges or diverges. If it converges, give the limit.

72, 18, 9 divided by 2, 9 divided by 8, ...   (5 points)

 

           

            Converges; 0

            Converges; -6120

            Diverges

            Converges; 96

 

13.       Find an explicit rule for the nth term of the sequence.

3, -12, 48, -192, ... (5 points)

 

           

            an = 3 • (-4)n - 1

            an = 3 • 4n + 1

            an = 3 • 4n - 1

            an = 3 • (-4)n

 

14.       Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are 40 and 5000, respectively. (5 points)

           

            an = 8 • 5n - 2

            an = 8 • 5n + 3

            an = 8 • 5n - 1

            an = 8 • 5n + 1

 

15.       Write the sum using summation notation, assuming the suggested pattern continues.

4 - 12 + 36 - 108 + ... (5 points)

 

           

            summation of 4 times negative 3 to the power of the quantity n minus 1 from n equals 0 to infinity

            summation of 4 times 3 to the power of the quantity n plus 1 from n equals 0 to infinity

            summation of 4 times 3 to the power of n from n equals 0 to infinity

            summation of 4 times negative 3 to the power of n from n equals 0 to infinity

 

16.       Find the standard form of the equation of the parabola with a focus at (0, -6) and a directrix at y = 6. (5 points)

           

            y2 = -24x

            y = negative 1 divided by 6x2

            y2 = -6x

            y = negative 1 divided by 24x2

 

17.       A tunnel is in the shape of a parabola. The maximum height is 16 m and it is 16 m wide at the base, as shown below.

 

What is the vertical clearance 7 m from the edge of the tunnel? (5 points)

 

           

            15.4 m

            0.3 m

            15.7 m

            0.6 m

 

18.       Find the center, vertices, and foci of the ellipse with equation x squared divided by 16 plus y squared divided by 25 = 1. (5 points)

           

            Center: (0, 0); Vertices: (0, -5), (0, 5); Foci: (0, -3), (0, 3)

            Center: (0, 0); Vertices: (-5, 0), (5, 0); Foci: (0, -4), (0, 4)

            Center: (0, 0); Vertices: (0, -5), (0, 5); Foci: (-4, 0), (4, 0)

            Center: (0, 0); Vertices: (-5, 0), (5, 0); Foci: (-3, 0), (3, 0)

 

19.       Find the center, vertices, and foci of the ellipse with equation 5x2 + 8y2 = 40. (5 points)

           

            Center: (0, 0); Vertices: (-8, 0), (8, 0); Foci: Ordered pair negative square root 39 comma 0 and ordered pair square root 39 comma 0

            Center: (0, 0); Vertices: (0, -8), (0, 8); Foci: Ordered pair 0 comma negative square root 39 and ordered pair 0 comma square root 39

            Center: (0, 0); Vertices:  Ordered pair 0 comma negative 2 square root 2 and ordered pair 0 comma 2 square root 2; Foci: Ordered pair 0 comma negative square root 3 and ordered pair 0 comma square root 3

            Center: (0, 0); Vertices: Ordered pair negative 2 square root 2 comma 0 and ordered pair 2 square root 2 comma 0; Foci: Ordered pair negative square root 3 comma 0 and ordered pair square root 3 comma 0

 

20.       A satellite is to be put into an elliptical orbit around a moon as shown below.

The moon is a sphere with radius of 567 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 900 km to 169 km. (5 points)

 

           

            x squared divided by 900 plus y squared divided by 169 = 1

            x squared divided by 1467 squared plus y squared divided by 736 squared = 1

            x squared divided by 736 squared plus y squared divided by 1467 = 1

            x squared divided by 169 plus y squared divided by 900 = 1

 

21.       Find the vertices and foci of the hyperbola with equation quantity x plus 4 squared divided by 9 minus the quantity of y plus 3 squared divided by 16 = 1. (5 points)

           

            Vertices: (-1, -3), (-7, -3); Foci: (-9, -3), (1, -3)

            Vertices: (-3, 0), (-3, -8); Foci: (-3, -8), (-3, 0)

            Vertices: (0, -3), (-8, -3); Foci: (-8, -3), (0, -3)

            Vertices: (-3, -1), (-3, -7); Foci: (-3, -9), (-3, 1)

 

22.       Find an equation in standard form for the hyperbola with vertices at (0, ±4) and asymptotes at y = ± 2 divided by 3.x. (5 points)

           

            y squared over 36 minus x squared over 16 = 1

            y squared over 16 minus x squared over 36 = 1

            y squared over 9 minus x squared over 4 = 1

            y squared over 16 minus x squared over 9 = 1

 

23.       Find all polar coordinates of point P where P = ordered pair 4 comma negative pi divided by 3. (5 points)

           

            (4, negative pi divided by 3 + 2nπ) or (-4, negative pi divided by 3 + (2n + 1)π)

            (4, negative pi divided by 3 + (2n + 1)π) or (-4, negative pi divided by 3 + 2nπ)

            (4, negative pi divided by 3 + 2nπ) or (4, pi divided by 3 + (2n + 1)π)

            (4, negative pi divided by 3 + 2nπ) or (-4, negative pi divided by 3 + 2nπ)

 

24.       Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = -1 - 3 sin θ (5 points)

 

           

            x-axis only

            Origin only

            y-axis only

            No symmetry

 

25.       Determine if the graph is symmetric about the x-axis, the y-axis, or the origin.

r = 4 cos 3θ (5 points)

 

           

            x-axis, y-axis

            x-axis only

            y-axis only

            x-axis, y-axis, origin

 

26.       Find the derivative of f(x) = negative 8 divided by x at x = 11. (5 points)

           

            8 divided by 121

            8 divided by 11

            121 divided by 8

            11 divided by 8

 

27.       Find the limit of the function by using direct substitution.(6 points)

limit as x approaches zero of quantity x squared plus ten. 

 

           

            -10

            Does not exist

            0

            10

 

28.       Find the limit of the function algebraically. (5 points)

limit as x approaches negative nine of quantity x squared minus eighty one divided by quantity x plus nine. 

 

           

            Does not exist

            1

            -9

            -18

 

 

29.       Use the given graph to determine the limit, if it exists.

 

 

 

Find  and . (5 points)

 

 

 

30.       Use graphs and tables to find the limit and identify any vertical asymptotes of the function. (5 points)

 

 

 

 

31.       Find the fifth roots of 243(cos 260° + i sin 260°). (5 points)

 

 

 

 

32.       The position of an object at time t is given by s(t) = 6 - 14t. Find the instantaneous velocity at t = 6 by finding the derivative. (5 points)

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