IB Stats Triangle trigonometry February 12, 2014
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1 I Stats Triangle trigonometry ebruary 12, The height of a vertical cliff is 450 m. The angle of elevation from a ship to the top of the cliff is 23. The ship is x metres from the bottom of the cliff. raw a diagram to show this information. alculate the value of x. 2. The diagram shows the plan of a playground with dimensions as shown. 48 m 117º 57 m alculate the length ; the area of triangle. 3. The diagram shows a water tower standing on horizontal ground. The height of the tower is 26.5 m. x m rom a point on the ground the angle of elevation to the top of the tower is 28. On the diagram, show and label the angle of elevation, 28. alculate, correct to the nearest metre, the distance x m. 4. The following diagram shows a carton in the shape of a cube 8 cm long on each side: H The longest rod that will fit on the bottom of the carton would go from to. ind the length l of this rod. ind the length L of the longest rod that would fit inside the carton.
2 5. gardener pegs out a rope, 19 metres long, to form a triangular flower bed as shown in this diagram. 5 m 6 m alculate the size of the angle ; the area of the flower bed. 6. n office tower is in the shape of a cuboid with a square base. The roof of the tower is in the shape of a square based right pyramid. The diagram shows the tower and its roof with dimensions indicated. The diagram is not drawn to scale. O H 10 m 40 m 6 m alculate, correct to three significant figures, (iii) (iv) the size of the angle between O and ; the shortest distance from O to ; the total surface area of the four triangular sections of the roof; the size of the angle between the slant height of the roof and the plane H; (v) the height of the tower from the base to O. parrot s nest is perched at a point, P, on the edge,, of the tower. person at the point, outside the building, measures the angle of elevation to point P to be 79. ind, correct to three significant figures, the height of the nest from the base of the tower. 2
3 7. rectangular block of wood with face leans against a vertical wall, as shown in the diagram below. = 8 cm, = 5 cm and angle  = 28. Wall 28º round ind the vertical height of above the ground. 8. is a trapezium with = and [] parallel to []. = 22 cm, = 12 cm, = 13 cm. Show that = 5 cm. alculate the height of the trapezium. (c) alculate Â; Ĉ. (d) alculate the length of the diagonal []. 9. In the diagram below, PQRS is the square base of a solid right pyramid with vertex V. The sides of the square are 8 cm, and the height V is 12 cm. M is the midpoint of [QR]. iagram not to scale V V = 12 cm P 8 cm M Q S 8 cm R Write down the length of [M]. alculate the length of [VM]. ind the total surface area of the pyramid; the angle between the face VQR and the base of the pyramid. 3
4 10. The diagram below shows a crane PQR that carries a flat box W. (PQ) is vertical, and the floor (PM) is horizontal. R 7.8 m 6.5 m Q 102º W 11.1 m h P M iven that PQ = 11.1m, QR = 7.8 m, P Qˆ R =102 and RW = 6.5 m, calculate PR; angle P Rˆ Q; (c) the height, h, of W above (PM). 11. ndrew is at point in a park. deer is 3 km directly north of ndrew, at point. rian is 1.8 km due west of ndrew, at point. (c) raw a diagram to represent this information. alculate the distance between rian and the deer. rian looks at ndrew, and then turns through an angle θ to look at the deer. alculate the value of θ. 12. Three right pyramids ndal, atsu and artos were discovered in the dense jungle of Marhartmasol. ach pyramid has a square base with centres, and respectively. ndal artos atsu iagram not to scale surveying team was lowered from a helicopter to the top of ndal to take measurements of the area. ndal is 40 metres high. The angle of elevation from the top of ndal to the top of atsu is 3. The horizontal distance from, the centre of the base of ndal, to, the centre of the base of atsu is 600 metres. Use the diagram below to find the height of atsu. 4
5 iagram not to scale 3º 40 m ndal atsu 600 m artos is found to be 92 metres high and the angle of elevation from the top of ndal to the top of artos is 4. raw a diagram similar to the diagram in part to show the relationship between ndal and artos. What is the horizontal distance from to? (c) The diagram below represents measurements relative to the centres of the bases of the pyramids. The surveyors determined the angle at to be 110, and the distance to be 600 m. iagram not to scale 110º 600 m (iii) What is the distance between and? ive your answer to the nearest metre. What is the size of angle? What is the area of the land inside triangle? 5
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