Math Practice Problems Module 4

1. Reduce (-3+2i)6 / 169 to the form rcis .

a. 13 cis 788051’43”b. 13 cis 877051’34”c. 13cis 787051’43”d. 13 cis 887051’34”

2. The trigonometric expression cos(arccosx) is equal to

a. 4x3+3xb. 4x3-3xc. 4x2+3xd. 4x2-3x

3. If arctan (1+x) + arctan (1-x) = arctan (1/2), find x.

a. 1b. 2c. 3d. 4

4. A tree stands vertically on a hillside that makes an angle of 220 with the horizontal. From a point 15 m down the hill from the base of the tree, the angle of elevation of the top of tree is 440. Find the distance from the point to the top of the flagpole.

a. 7.51 mb. 7.61 mc. 7.71 md. 7.81 m

5. A flagpole 49 m high is situated at the top of a hill. At a point 183 m down the hill, the angle between the surface of the hill and line to the top of the flagpole is 8 degrees. Find the distance from the point to the top of the flagpole.

a. 223.1 mb. 232.1 mc. 322.1 md. 233.1 m

6. The angle of depression of a boat observed from the top of a lighthouse is 3028’. If the lighthouse is 27 m high, find the distance of the boat from the base of the lighthouse.

a. 445.30 mb. 445.50 mc. 445.70 md. 445.90 m

7. Two ships left the same port, one going in the direction N700E and the other sailing directly east. The first ship traveled at the rate of 12 kilometers per hour, After 30 minutes, the second ship was observed to the directly south of the first. Find the speed of the second ship.

a. 10.82 km/hrb. 12.28 km/hrc. 13.82 km/hrd. 11.28 km/hr

8. A pole stands on an inclined plane, which makes an angle of 38021’ with the horizontal. At the base of the inclined plane, the pole subtends an angle of 19036’; at a point 11 meters up the inclined plane, it subtends an angle of 48032’. Find the height of the pole.

a. 7.29 mb. 7.39 mc. 7.49 md. 7.59 m

9. A 90-degree arc on the earth is equal to how many nautical miles in length?

a. 5, 600b. 5, 400c. 5, 200d. 5, 000

10. The plane of a small circle on a sphere of radius 25 cm is 7 cm from the center of the sphere. Find the radius of the small circle.

a. 21b. 22c. 23d. 24

11. Find in kilometers the length of an arc of a great circle on the earth if its length is 12.5 degrees.

a. 1389b. 1370c. 1352d. 1333

12. A spherical triangle has angles A, B and C, each of which is less than 180 degrees. Which of the following is true?

a. A + B + C = 1800

b. A + B + C > 3600

c. A + B + C = 3600

d. A + B + C > 1800

13. Find the difference in longitude between two points A and B if A is in longitudes 450 E and B in longitude 1550W.

a. 1100

b. 1600

c. 1800

d. 2000

14. You are in latitude 400N. How far are you from the equator in statute miles?

a. 2, 476.8b. 2, 674.8c. 2, 764.8d. 2, 467.8

15. If the radius of the earth is 6, 370 km, find the radius of a parallel of latitude 60 degrees north.

a. 3, 185 kmb. 3, 158 kmc. 3, 518 kmd. 3, 581 km

16. Find the distance in nautical miles between A (40030’N,600E) and B (80020’S,600E).

a. 5,270b. 5,720c. 7,520d. 7250

17. Find the difference in longitude between New York (40043’N, 740W) and Sydney (33052’S, 151013’E).

a. 77013’b. 134047’c. 134087’d. 77031’

18. If A is 720 nautical miles south of the equator, find the latitude of A in degrees.

a. 140Sb. 120Sc. 100Sd. 130S

19. Find the area of a spherical triangle with angles 34023’, 119037’and 38043’on a sphere of radius 10.

a. 24.19b. 23.19c. 22.19d. 21.19

20. A spherical triangle, which contains at least one side equal to a right angle, is called

a. a polar triangleb. a right trianglec. an isosceles triangled. a quadrantal triangle

21. A ship sails in latitude 320N due east until it has made good a difference in longitude of 2035’. Find the departure.

a. 131.45 nmb. 141.35 nmc. 135.41 nmd. 145.31 nm

22. Using Napier’s Rule, write a formula to find angle A when angle B and side c are given.

a. tanA = cosc tanBb. cotA = cosc tanBc. tanA = tanc cosBd. cotA = tanc cosB

23. In a right spherical triangle whose angles are A = 63015’, B = 135034’ and C = 900, find side b.

a. 134.10

b. 143.10

c. 131.40

d. 141.30

24. A ship leaves M (360N, 760W) and sailing on a great circle arc crosses the equator at 500W. Find the distance traveled.

a. 2,60l nmb. 2,501 nmc. 2,401 nmd. 2,301 nm

25. Given a spherical triangle with A = 74021’, B = 83041’ and C = 58039’. Find c.

a. 54.930

b. 56.800

c. 55.730

d. 57.360

26. Given: a = 51031’, b = 36047’ and c = 80012’. Find A

a. 34.450

b. 34.550

c. 34035’d. 34025’

27. Given: a = 68027’, b = 87032’ and C = 97053’. Find c.

a. 94.410

b. 97.410

c. 96.410

d. 95.410

28. How long does it take to sail from Manila (14036’N, 12105’E) to Hong Kong (22018’N, 114010’E) at the rate of 18 mph?

a. 1.5 daysb. 1.6 daysc. 1.7 daysd. 1.8 days

29. Express in hour, minutes and seconds the time corresponding to 260034’.a. 17h20m16sb. 17h21m16sc. 17h22m16sd. 17h23m16s30. In a quadrantal spherical triangle where a = 540, b = 380 and c = 900, find angle A.

a. 16.310

b. 17.310

c. 18.310

d. 19.310

31. An icosahedron is a regular polyhedron having

a. 12 facesb. 16 facesc. 18 facesd. 20 faces

32. The perpendicular distance from the center of a regular polygon to a side is called the_________ of the polygon.

a. altitudeb. amplitudec. mediand. apothem

33. The angle formed by two intersecting planes is called a

a. face angleb. vertical anglec. dihedral angled. central angle

34. Find the area of a pentagon whose apothem is 10 cm.

a. 336.72 cm2

b. 373.65 cm2

c. 327.36 cm2

d. 363.27 cm2

35. The radii of two spheres are in the ratio 3:4 and the sum of their surface area is 2,500 sq. cm. Find the radius of the smaller sphere.

a. 15 cmb. 10 cmc. 20 cmd. 25 cm

36. The base of a pyramid is a regular hexagon whose apothem is 8.66 cm. If the altitude of the pyramid is 6 cm, find the volume of the pyramid rounded to 3 significant figures.

a. 510 cu. Cmb. 515 cu. Cmc. 520 cu. Cmd. 525 cu. Cm

37. A face diagonal of a cube is 4 cm. Find the volume of the cube.

a. 22.63 cm3

b. 21.64 cm3

c. 23.62 cm3

d. 24.64 cm3

38. What is the distance in cm between two vertices of a cube, which are farthest from each other, if an edge measures 5 cm?

a. 52b. 53c. 54d. 55

39. The volume of a cube is reduced to _____ if all sides are halved.

a. ½b. ¼c. 1/8d. 1/16

40. The sum of the angles of a polygon of n sides equals.

a. 3600

b. 1800

c. (n-2) x1800

d. (n-2)(1800)   / n

41. Each exterior angles of a polygon of n sides is

a. 180/nb. 360/nc. (n-2)(1800)  / nd. always 720

42. The median of a trapezoid equals _________ the sum of the bases.

a. 1/3b. ½c. ¼d. 1/5

43. A pyramid has a base whose sides are 10 m, 16 m and 18 m. if the altitude of the pyramid is 20 m, find the volume of the inscribed cone in m3.

a. 274.18b. 1715.36c. 325.1d. 376.57

44. The tangent and a secant are drawn to a circle from the same external point. If the tangent is 6 inches and the external segment of   the secant is 3 inches, the length of the secant is _____inches.

a. 12b. 15c. 14d. 18

45. Two secants from a point outside the circles are 24 and 32. If the external segment of the first is 8, the external of the second is:

a. 8b. 6c. 10d. 12

46. A trench is constructed so that its cross section is a trapezoid the area of which is 21 square feet. If the shorter base is 2 times its height and the longer base is 5 ft longer than its height, find the height of the trench.

a. 3 ftb. 4 ftc. 5 ftd. 6 ft

47. The second proposition of Pappus states that the volume of a solid of revolution is equal to the generating area times the circumference of the circle described by the centroid of the area. Find the volume of the solid formed by revolving a circle around a line tangent to it. Let R be the radius of the circle.

a. 19.74 R3

b. 100R3

c. 18.85 R3

d. R3

48. If a circular cylindrical tank axis horizontal, diameter 1 meter, and length 2 meters is filled with water up to a depth of 0.75 meter, how much water is in the tank?

a. 2 m3

b. 1.51 m3

c. 1.2637 m3

d. 3.1415 m3

49. The diagonals of a rhombus are equal to 30 cm and 40 cm. Find the distance between the plane of the rhombus and a point M if the latter is 20 cm distant from each of its sides.

a. 15 cmb. 16 cmc. 17 cmd. 18 cm

50. The angle at the vertex of an axial section of a cone is a right one. The area of the section is equal to 25 cm2. Find the area of its base.

a. 25 cm2

b. 30 cm2

c. 50 cm2

d. 625 cm2