Applications of trigonometry Problem set B

The cuboid ABCDEFGH has AB = 7 cm, BC = 6 cm and AE = 4 cm.

1. (a) Find the length of AG.
2. (b) Find the angle between AG and the base ABCD.
3. (c) Find the angle between AG and the edge AB.
4. (d) AG and CE cross at the centre of the cuboid. Find the acute angle between AG and CE.

The cuboid ABCDEFGH has AB = 13 cm, BC = 5 cm and AE = 6 cm.

1. (a) Find AC, then find the angle ACE.
2. (b) Find the length of HB.
3. (c) Find the angle between HB and BD.
4. (d) Find the area of the triangle BGE.

In the cuboid ABCDEFGH the three face diagonals AC, AF and CF have lengths AC = 12 cm, AF = 9 cm and CF = 10 cm.

1. (a) Find the length of the diagonal AG. Give your answer to 4 significant figures.
2. (b) Find the angle between AG and the base ABCD.

EABCD is a right pyramid with a square base ABCD of side 6 cm. Each sloping edge is 9 cm long. M is the midpoint of AB and N is the midpoint of BC.

1. (a) Sketch the triangle ACE and label every length you can.
2. (b) Find the height of the pyramid.
3. (c) Find the angle AEC.
4. (d) Find the angle MEN.

MABCD is a right pyramid. The base ABCD is a rectangle with AB = 12 cm and BC = 5 cm. O is the centre of the base and OM = 8 cm is the height.

1. (a) Find the angle ACM, which is the angle between the sloping edge CM and the base.
2. (b) Find the angle ABM.

Two solids.

1. (a) MABC is a right pyramid. The base ABC is an equilateral triangle of side 8 cm. O is the centre of the base and OM = 6 cm. Find the angle between AM and the base.
2. (b) A cone has a base of diameter 16 cm and a vertical height of 10 cm. Find the angle between a sloping edge and the base.

A hiker leaves a campsite C and walks 7.4 km on a bearing of 285° to a lake L. She then walks 5.2 km on a bearing of 210° to a hut H.

1. (a) Find the distance from H back to C.
2. (b) Find the bearing on which it must travel from H to return to C. Give the bearing to the nearest degree.

From a radar station R, a ship S is 5.2 km away on a bearing of 310° and a buoy B is 3.9 km away on a bearing of 047°.

1. (a) Find the distance from the ship to the buoy.
2. (b) Find the bearing of the ship from the buoy. Give the bearing to the nearest degree.

A campsite C has a flagpole F 310 m away on a bearing of 308°. A stream S is due west of the campsite and due south of the flagpole.

1. (a) Sketch the situation. Mark the north direction at the starting point, the bearing and the right angle.
2. (b) Find the distance from the stream to the flagpole.

A drone flies 5.8 km on a bearing of 072°, then 2.9 km on a bearing of 158°, then 3.6 km on a bearing of 252°. Find how far it is from where it took off.

Find the answer to each part. Draw a sketch with a right-angled triangle first.

1. (a) A surveyor's eyes are 1.65 m above the ground. She stands 22 m from a tree and sees the top at an angle of elevation of 29°. Find the height of the tree.
2. (b) From a point on level ground 42 m from the foot of a cliff, the angle of elevation of the top of the cliff is 33°. Find the height of the cliff.
3. (c) A kite is 45 m above the ground. It is seen at an angle of elevation of 38° from a point on level ground. Find the horizontal distance from the point to the kite.
4. (d) From the top of a cliff 85 m high, the angle of depression of a boat is 23°. Find the horizontal distance from the foot of the cliff to the boat.

Three problems about heights and angles.

1. (a) A lamp-post stands on level ground. From a point on the ground the angle of elevation of the top is 41°. After walking 6 m directly away from the lamp-post, the angle of elevation is 30°. Find the height of the lamp-post.
2. (b) From the top of a ski-lift tower, the angle of depression of a skier due east of the tower is 27.0°. A second skier is 30 m due south of the first. Her angle of depression from the top is 25.0°. Find the height of the tower.
3. (c) A flagpole stands at B on level ground. Ana stands at P, 31.5 m from B, and Sam stands at Q, 22.8 m from B. They are 11.9 m apart. Find the angle PBQ.

A shopping centre has an entrance in the shape of a glass square-based pyramid. The base has sides of 24 m and the height is 13 m. Heating is rated at one unit for every 900 m³ of air.

1. (a) How many units are needed? State your assumptions.
2. (b) On a cold day the glass loses energy at 210 watts for every m² of glass. Find the total power the heating must supply. The base is not glass.
3. (c) The cleaning rules say scaffolding is needed if any glass has an angle of elevation of more than 50°. Is scaffolding needed?

A tent has a floor 3.8 m long and 2.6 m wide. Its roof is a triangular prism with an isosceles triangle as its end. Each side of the tent makes an angle of 62° with the ground. A person can stand only where the tent is at least 0.9 m high.

1. (a) Find the height of the ridge above the floor.
2. (b) Find the volume of air inside the tent.
3. (c) Find the percentage of the floor area where the tent is at least 0.9 m high.
4. (d) Find the percentage of the volume of air that is at least 0.9 m above the floor.