5.8 Trigonometry 3D problems
Chapter
Chapter 5
Section
5.8
Solutions 17 Videos

Mina is trolling for salmon in Lake Ontario. She sets the fishing rod so that its tip is 1 m above water and the line forms an angle of 35^o with the water’s surface. She knows that there are fish at a depth of 45 m. Describe the steps you would use to calculate the length of line she must let out.

1.47mins
Q1

Josh is building a garden shed that is 4.0 m wide. The two sides of the roof are equal in length and must meet at an angle of 80^o. There will be a 0.5 m overhang on each side of the shed. Josh wants to determine the length of each side of the roof.

a) Should he use the sine law or the cosine law? Explain.

b) How could Josh use the primary trigonometric ratios to calculate x?

Explain.

1.42mins
Q2

Determine the value of x to the nearest centimetre. Explain your reasoning for each step of your solution.

1.12mins
Q3a

Determine the value of x to the nearest centimetre and \theta to the nearest degree. Explain your reasoning for each step of your solution.

2.06mins
Q3b

Determine the value of x to the nearest centimetre and \theta to the nearest degree. Explain your reasoning for each step of your solution.

3.04mins
Q3c

Determine the value of x to the nearest centimetre and \theta to the nearest degree. Explain your reasoning for each step of your solution.

3.02mins
Q3d

As a project, a group of students was asked to determine the altitude, h, of a promotional blimp. The students’ measurements are shown in the sketch at the left.

a) Determine h to the nearest tenth of a metre. Explain each of your steps.

b) Is there another way to solve this problem? Explain.

2.07mins
Q4

While Trey and Bill were flying a hot-air balloon from Beamsville to Vineland in southwestern Ontario, they decided to calculate the straight-line distance, to the nearest metre, between the two towns.

• From an altitude of 226 m, they simultaneously measured the angle of depression to Beamsville as 2^{\circ} and to Vineland as 3^{\circ}.
• They measured the angle between the lines of sight to the two towns as 80^{\circ}.

Is there enough information to calculate the distance between the two towns? Justify your reasoning with calculations.

4.32mins
Q5

The observation deck of the Skylon Tower in Niagara Falls, Ontario, is 166 m above the Niagara River. A tourist in the observation deck notices two boats on the water. From the tourist's position,

• the bearing of boat A is 180^{\circ} at an angle of depression of 40^{\circ}.
• the bearing of boat B is 250^{\circ} at an angle of depression of 34^{\circ}.

Calculate the distance between the two boats to the nearest metre.

4.47mins
Q6

Suppose Romeo is serenading Juliet while she is on her balcony. Romeo is facing north and sees the balcony at an angle of elevation of 20^o. Paris, Juliet’s other suitor, is observing the situation and is facing west. Paris sees the balcony at an angle of elevation of 18^o. Romeo and Paris are 100 m apart as shown. Determine the height ofJuliet’s balcony above the ground, to the nearest metre.

3.41mins
Q7

A coast guard helicopter hovers between an island and a damaged sailboat.

• From the island, the angle of elevation to the helicopter is 73^o.
• From the helicopter, the island and the sailboat are 40^o apart.
• A police rescue boat heading toward the sailboat is 800 m away from the scene of the accident. From this position, the angle between the island and the sailboat is 35^o.
• At the same moment, an observer on the island notices that the sailboat and police rescue boat are 68^o apart.

Explain how you would calculate the straight-line distance, to the nearest metre, from the helicopter to the sailboat. Justify your reasoning with calculations.

2.47mins
Q8

Briana and Tanya are standing 8.8 m apart on a dock when they observe a sailboat moving parallel to the dock. When the boat is equidistant between both girls, the angle of elevation to the top of its 8.0 m mast is 51^{\circ} for both observers. Describe how you would calculate the angle, to the nearest degree, between Tanya and the boat as viewed from Briana's position. Justify your reasoning with calculations.

3.25mins
Q9

Bert wants to calculate the height of a tree on the opposite bank of a river. To do this, he lays out a baseline 80 m long and measures the angles as shown at the left. The angle of elevation from A to the top of the tree is 28^o. Explain if this information helps Bert to calculate the height of the tree to the nearest metre. Justify your reasoning with calculations.

1.42mins
Q11

Channy's homework question reads like this: Bill and Chris live at different intersections on the same street, which runs north to south. When both of them stand at their front doors, they see a hot-air balloon toward the east at angles of elevation of 41^o and 55^o, respectively. Calculate the distance between the two friends.

a) Channy says she doesn’t have enough information to answer the question. Evaluate Channy's statement. Justify your reasoning with calculations.

b) What additional information, if any, would you need to solve the problem? Justify your answer.

1.44mins
Q12

Two roads intersect at 34^o. Two cars leave the intersection on different roads at speeds of 80 km/h and 100 km/h. After 2 h, a traffic helicopter that is above and between the two cars takes readings on them. The angle of depression to the slower car is 20^o, and the straight—line distance from the helicopter to that car is 100 km. Assume that both cars are travelling at constant speed.

• Calculate the straight—line distance, to the nearest kilometre, from the helicopter to the faster car. Explain your reasoning for each step of your solution.
2.20mins
Q13a

Two roads intersect at 34^o. Two cars leave the intersection on different roads at speeds of 80 km/h and 100 km/h. After 2 h, a traffic helicopter that is above and between the two cars takes readings on them. The angle of depression to the slower car is 20^o, and the straight—line distance from the helicopter to that car is 100 km. Assume that both cars are travelling at constant speed.

• Determine the altitude of the helicopter to the nearest kilometre.