b) At age 16 and older, you can get a driver s license. c) Most cars today last less than 200,000 miles.

Transcription

1 Unit 3 Geometry in Construction Name Block 1. Write the tolerances/ranges shown below as an inequality statement (algebraically). Graph on a number line. a) Below 21 years of age, you have a probationary driver s license. b) At age 16 and older, you can get a driver s license. c) Most cars today last less than 200,000 miles. d) All the numbers less than -6. e) An angle in a triangle must be smaller than 180. f) We have up to but no more than 22 days to finish the framing. 2. Laura drew the triangle shown below. Side a is 7.8 cm in length. Side c is 9.5 cm in length. What is the measure of angle x, in degrees? Page 1 Contextual Learning Concepts, LLC Copyright 2013 No

2 3. Mr. Burke measured one of the spotlights that shines on the Washington Monument in Washington, DC which beams its light 630 feet to the top of the monument. The spotlight is 290 feet from the center of the base of the monument, as shown. What angle (x), in degrees, does the beam of light from the spotlight make with the ground? Round to the nearest degree. 4. A flagpole, which is 20 feet high, is positioned on top of a two-story school that has a flat roof. The roof of the school is 33 feet from the ground. The distance from the top of the flagpole to the front of the parking lot is 255 feet. What is the angle of elevation (x) from the front of the parking lot to the top of the flagpole? 5. Find the angle (x) that each stud needs to be cut to create the required cathedral wall shown in the wall diagram below. Page 2

3 6. The parking lot shown below is to be tripled in size. If the original lot takes 5 hours of time to pave the lot, how much time will the new larger lot require? 7. Solve a) 3x 5 14 b) 3x 5 14 c) 2h 3 11 d) 2h 3 11 e) 5y 2 37 f) 5y According to the parking standards in Loveland, an access ramp to a parking lot cannot have a slope exceeding10. Suppose a parking lot is 11 feet above the road. If the length of the ramp is 55 ft., does this access ramp meet the requirements of the code? Explain by showing the math problem. 9. Find the distance across the pond. Page 3

4 10. There are two ways of examining trusses by slope and by degrees. a. Truss B outlines two right triangles. To the nearest tenth of a degree, what is the size of each angle in these right triangles? b. Suppose the vertical center post in Truss B is 5 feet long. Find the lengths of the other two sides of each right triangle. Round your answer to the nearest inch. c. The top part of Truss C outlines two right triangles. To the nearest tenth of a degree, what is the size of each angle in these right triangles? d. Suppose the vertical center post in Truss C is 9 feet long. Find the lengths of the other two sides of each right triangle. Round your answer to the nearest inch. 11. A wheelchair ramp makes a 4 angle with the ground. The beginning of the ramp is 360 inches from the end of the ramp. What is the length of the ramp to the nearest inch? Page 4

5 12. Commercial aircraft usually fly at an altitude between 29,000 and 36,000 feet. When landing, their gradual descent to an airport runway occurs over a long distance. a. Suppose a commercial airliner flying at 29,000 feet begins its descent with an angle of descent of 2.5. At what horizontal distance from the runway should the descent begin? b. Suppose a commercial airliner flying at an altitude of 36,000 feet begins its descent at a horizontal distance 180 miles from the runway. What is the angle of descent? 13. Each hip rafter shown rises to the top of the roof at a 30. How long is a hip rafter? 14. The world s tallest unsupported flagpole is a 282 feet tall steel pole in Surrey, British Columbia. The shortest shadow cast by the pole during the year is 137 ft. To the nearest degree, what is the angle of elevation of the sun when the shortest shadow is cast? 15. The Americans with Disabilities Act states that wheelchair ramps can have a slope no greater than 1. Find the angle of elevation of a ramp with this slope. Round your 12 answer to the nearest tenth. Page 5

6 16. Solve and then graph on the number line. a) 3v 2 17 b) 5t 7 3 c) 8 4g 20 d) x e) 6 3r 9 f) 7b To guard against a fall, a ladder should make an angle of 75 degrees or less with the ground. What is the maximum height that a 20 ft. ladder can reach safely? What is the maximum height a 16 ft. ladder can reach safely? 18. Two office buildings are 160 ft. apart. The height of the taller building is 635 ft. The angle of depression from the top of the taller building to the top of the shorter building is 15. Find the height of the shorter building to the nearest foot. Page 6

7 19. A candy company is doing a sales promotion for a new flavor of candy. The candy is sold in a small wrapper. On the inside of each wrapper, one of 5 different prizes is shown. The prizes are listed equally and randomly on the wrappers of the new flavored candy. If two packages of the candy are purchased, what is the probability that the prizes listed on the inside of the lid will be different? Hint: List out the outcomes 20. All but two of the pyramids built by the ancient Egyptians have faces inclined at 52 angles. Suppose you found the ruins of a pyramid. Most of the pyramid has eroded, but you are able to determine that the length of a side of the square base is 262 ft. How tall was the pyramid, assuming its faces were inclined at 52? Round your answer to the nearest foot. 21. A flagpole in Denver is supported by two wires, as shown below. What is the length, to the nearest hundredth of a yard, of each of the flagpole s support wires? Page 7

8 22. An airplane that is taking off makes a 13 angle of elevation from the runway. The airplane, shown below, is directly above the point on the runway that is 1,800 feet from the airplane s take-off point. At what height (h) is the airplane when it is directly above the point that is 1,800 feet from its take-off point? Round your answer to the nearest foot. 23. The grade of a road is a percent calculated from the ratio Vertical distance traveled Horizontal distance traveled The road in the sketch below has a 5% grade. a) Find the angle of incline (x) of the road. Hint: 5% means 5 over what? b) A very steep street has a grade of 15%. If you travel 1000 feet horizontally, how much has your elevation changed? Page 8

9 24. The angle of elevation is the angle between the horizontal and the line of sight. The angle of elevation of the roof of this building is 30. Find the height of the building. 25. Find the missing measure. Round to the nearest tenth. 26. Mr. Brown is building a wooden ramp to allow people who use wheelchairs easier access to the public library. The ramp must be 2 feet tall. Find the angle of elevation if the ramp begins 24 feet away from the library. Round to the nearest tenth. 27. Felicia has 120 m of electric fencing. She wants to enclose a rectangle of the back yard having the maximum possible area. What is the design and maximum area that can be enclosed? Give exact measurements.no decimals. 28. Arrange these numbers in order from smallest to largest. 3.0 x x x x 10-2 Page 9

10 29. The parking lot shown below is to be tripled in dimension. If the original lot takes 5 hours of time to fence the lot, how much time will the new larger lot require? 30. Which of these represents the largest value? (5 5) Simplify a) x x b) 2 3 x x c) 2 3 x x Solve and graph on a number line. a) 1.6b b) g c) 2( b 3) 4( b 6) d) 7x 4 3x 16 Page 10

11 33. The Canadian National Tower in Toronto, Ontario, is approximately 1815 feet tall. This tower is the tallest free-standing structure in the world. a. At some time on a sunny day, the sun makes the tower cast a 600 ft. shadow. What is the measure of the angle formed by a sun ray and the ground at the tip of the shadow? b. From the top of the Canadian National Tower, a boat is observed in Lake Ontario, approximately 1.8 miles away from the base of the tower. Assume the base of the tower is approximately level with the lake surface. What angle below the horizontal must the observer look to see the boat? 34. A forest ranger sights a tree through a surveying instrument. The angle of elevation to the top of the tree is 26. The instrument is 4 feet above the ground. The surveyor is 110 feet from the base of the tree. To the nearest foot, how tall is the tree? 35. The following sketch shows the start of one surveyor s attempt to determine the height of a tall mountain without climbing to the top herself. How tall is the mountain? Page 11

12 36. An exit ramp for I-25 is to be built with a 4.34% grade to an overpass that is 33 feet above the road surface. How far horizontally from the overpass should the engineers plan to begin the ramp? 37. The Pyramid of Cheops in Egypt is a right square pyramid. The base edge measures about 754 feet and each face makes an angle with the horizontal desert floor of Determine the height of the pyramid. 38. A radio tower is anchored to the ground by four cables, two of which are shown in the figure shown below. Each cable is bolted to the tower 50 feet above the ground. The angles formed by the cables with the ground measure 60. a) How many feet of cable have been used to anchor the tower (remember 4 cables)? b) How far is it from the bottom of the tower to the anchor point of each cable? Page 12

13 39. Find the volume and surface area of each home. a) hemisphere b) regular hexagon 40. If a roof has a pitch of 5/12, then the rise is 5 and the run is 12. Determine the measure of the angle the roof in the diagram makes with the horizontal. a) 3/12 b) 8/12 c) 4/12 d) 6/12 e) What are the advantages or disadvantages of the various pitches? Page 13

14 41. Graph 2 y x Solve and graph on the number line a) 2x 30 b) 7 2x c) x d) x 43. A triangular roof truss has a total length of 46. The angle the truss rises is 45. Find the distance from the end of the truss to the peak. Page 14

15 44. A wheel chair ramp has a length of 36 and rises 12. Find the slope of the ramp. Find the angle of elevation? Is this an acceptable ramp? 45. The distance the bottom of a ladder should be safely placed away from a wall is 1/4 th of the height that the top of the ladder reaches. What is the angle of elevation of the ladder? 46. If a triangular truss height is 15 to the peak and the pitch of the roof is 6/12, find the distance from the end of the truss to the peak. 47. A ramp is 20 long. If we use this ramp to span a distance of 15, find the angle of elevation of the ramp. 48. Calculate the additional pipe (travel length) that is needed to keep two pipes parallel when they turn at a 72 angle. The pipes are 24 apart (offset). Make a sketch of the pipes and show where the measurements are located. An example is shown. 49. Calculate the additional pipe that is needed to keep two pipes parallel when they turn at an 70 angle. The pipes are 18 apart (offset). Make a sketch of the pipes and show where the measurements are located. Page 15

16 50. In this enlarged drawing of a bolt, the angle has a measure of 60 and a distance from thread to thread is 12 mm. Find the depth of each thread. 51. Three lengths of sheet metal, each with a width of 12 inches, are folded to make pipes. Their cross sections are shown here. Which of the pipes will allow the greatest flow of water? 52. Find the truss support length. Page 16

17 53. Imagine that you work for a construction company. Your construction firm has been hired to help develop the Eagle River Estates. A map of the property is shown below. a) Your company must build a bridge from Fox Corner to Dale. Determine the length of the bridge. b) After the bridge is completed, the company must install water pipe from Waverly and Eagle to Dale. Find the distance from Waverly to Dale and Eagle to Dale. 54. A swimming pool is 26 feet long and 22 feet wide. The shallow 2 foot deep end extends for 6 feet. Then for 14 feet horizontally, there is a constant decline toward the 8 foot deep end. A sketch from the side is shown. a) How much water is needed to fill the pool within 6 inches of the top? Assume 7.68 gallons per cubic foot. b) One gallon of paint covers 80 sq feet of surface. How many gallons of paint are needed to paint the inside of the pool? Page 17

18 55. Find the surface area (ignore floor but count the roof) and the volume of the home shown below. 56. Soil that is disturbed (dug up) will not be as compact as the soil is naturally. When soil is disturbed, it expands about 15%. If you are digging up 100 cubic yards of soil and hauling it away, how many truck loads will be needed to haul the soil away. Assume a truck hauls 15 yards. 57. A scale model of the Big Thompson Water pipe is 4 inches wide. If the actual pipe is 4 feet wide, what is the scale of the model? 58. Solve a) 2 6(3x 6) 18 b) 15x ( x 2) 59. Solve and graph a) 2 6(3x 6) 18 b) 15x ( x 2) 60. Find the slope of a line passing through (3,-7) and (-9,12) Page 18

19 61. Suppose that two rangers spot a fire from their fire towers, A and B. Use what you know about angles in a triangle and the Law of Sines to find the distances from each tower to the fire. 62. A commuter airplane, off course over the ocean, reported experiencing a mechanical problem. The pilot sent two calls, one to Anchorage International Airport and one to the airport in Chugiak. Air traffic controllers at the two airports reported the angles shown in the diagram below. How far was the plane from the closer airport? 63. Find the m C. Round to the nearest hundredth. Page 19

20 64. Find the length of NM. Round to the nearest hundredth. 65. Find the length of each indicated side to the nearest centimeter. a) b) c) 66. Find the measure of each angle A in each acute triangle to the nearest degree. a) b) c) Page 20

21 67. Find the perimeter of an isosceles triangle truss with a base of 25 feet, and a vertex angle of 35. Round to the nearest hundredth. 68. Find the m C if a = 9, b = 20, m B 105. Round to the nearest hundredth. 69. Solve the triangle (find all the missing measures). Round to the nearest hundredth. 70. Find the length of BC. Round to the nearest hundredth. 71. Find the length of b. Round to the nearest hundredth. Page 21

22 72. A metal brace is built with the design below. Measures are in inches. Find the length of a. Round to the nearest hundredth. 73. Find the perimeter of the triangle. Round to the nearest hundredth. 74. Find the m H. Round to the nearest hundredth. 75. If the drawing below was a truss, what would be the length of HJ? Round to the nearest hundredth. Page 22

23 76. A lot is shown below. The measures are in yards. Find the measure of Y. Round to the nearest hundredth. 77. Find the perimeter of the triangle. Round to the nearest hundredth. 78. Find the length of the missing side. Round to the nearest hundredth. j = 43, h = 53, m K A triangular bike frame has dimensions shown below in inches. Find the length of the missing side. Round to the nearest hundredth. a =25, b = 21, m C Find the area of the triangle. Round to the nearest hundredth. Page 23

24 81. Find the area of the triangular lot to the nearest square foot. 82. Find the area of the triangle. Round to the nearest hundredth. 83. Find the area of the triangular lot to the nearest square foot. 84. Find the area of the triangle. Round to the nearest hundredth. Page 24

25 85. Find the area of each of the triangles. Round to the nearest hundredth. a) b) 86. A construction company owns a triangular lot in at a Y intersection. The length of the property line AB is 140 feet. The length of the property line BC is 220 feet. The angle between the property lines AB and BC is 125. What is the area of the lot? Round to the nearest hundredth. 87. Find the m A given a = 31, b = 35, c = 44. Round to the nearest hundredth. 88. Find the m P. Round to the nearest hundredth. Page 25

26 89. Write the tolerances/ranges shown below as an inequality statement (algebraically). Graph on a number line. a) Two sides of a triangle are 6 and 8 inches long. What is the range of the 3 rd side? b) When cutting plumbing pipe, a 48 pipe can be no longer than 48.5 and no shorter than 47.5 c) A millionaire is considered anyone having at least 1 million and no more than 999 million. d) Cabins built similar to this year s cabin range in price from $58,000 to $86,000. e) During rough framing, a 86 wall can vary by 1/8 f) All numbers between -5 and 7 including -5 and 7. g) Construction workers typically earn between $10 and $15 per hour. Page 26

27 90. Sara is building a triangular pen for the pets. If two of the sides measure 7 feet and 10 feet, what is the range of lengths for the third side? 91. Logan has 2 boards measuring 34 inches and 56 inches. What is the range of the lengths for the third side if he is wanting to build a triangle? Write algebraically. 92. Given sides of a triangle measure 14 feet and 23 feet. What is the possible range of lengths for the third side write algegraically. 93. Solve and graph a) 0 5x 30 b) 15 6g What is the slope of a line passing through (90,-84) and (56,-84)? 95. What is the slope of a line perpendicular to #94? Page 27

28 96. The code requires that step risers (height) must be in between Express as an inequality. 3 6 and 7 4 inches (inclusive). 97. The standard set of steps have a 7 inch rise and a 11 inch tread (run). What is the angle of elevation? 98. Find the perimeter of the figure below. 99. The plan below shows the bay window floor section of a cabin. What is the angle that needs to be cut? Page 28

29 100. The Cooke family has purchased the lot adjoining their original property as shown in the map below. a) Calculate the length of side AB b) Find the area of each triangular lot Aaron is directly over the 2500 meter landing strip in his hot air balloon (point A) and is observed at one end of the strip by Betsy (Point B) with an angle of elevation measuring 65. Meanwhile, at the other end of the strip, Caleb (Point C) observes the balloon with an angle of elevation of 38. What is the distance between Aaron and Betsy? How high up is Aaron when he is observed? Page 29

30 102. A roof design is given below. a) What is the roof width (w)? b) Given the roof is 36 feet long, what is the roof s total surface area? c) What is the perpendicular height of the roof (n)? d) How would your answer to c above change if the angle of the roof changed from 28 to 35? 103. The window dimensions in a commercial building are being enlarged using a ratio of 2:3. If the total window area is 500 sq feet currently, what will the area of the new windows be? 104. What is the decimal that you would need to enter into a calculator for a measurement of 4 feet 5 inches? 105. Your calculator gives a measurement of 6.3 feet. What is the measurement you should use on your tape measure? Page 30