Question

2. A ski resort is building a new ski lift that will transport tourists from the base of the mountain to its highest point. This mountain has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. When the project is completed, how many yards, d, will a tourist travel from the base of the mountain to its peak?
Part I: Sketch a figure to illustrate the scenario above. Label the vertices and the lengths that are given in the question. (3 points)


Part II: Using your sketch from Part I, write an equation using a trigonometric ratio to find the distance a tourist will travel from the base of the mountain to its peak. Round your answer to the nearest 100th. Show your work. (2 points)


3. Darcy mounted a motion sensor so it would light a path to the door on her deck. If you know AB = 10 feet, and BE and BD trisect ∠ABC, what is the perimeter of the deck area to the right of the beam of light (△BDC)?

Part 1: What other angles or sides of △BDC can you label given that side AB is 10 feet, BE and BD trisect ∠ABC? Label the diagram accordingly, and explain your reasoning. (4 points)


Part 2: Use the trigonometric ratios 30° and 60° to calculate and label the remaining sides of △BDC. Show your work. (3 points)


Part 3: Use the Pythagorean Theorem to calculate the length of side BD. Does this method verify the length you found using trigonometric ratios? (2 points)


Part 4: What is the perimeter of the area to the right of the beam of light on Darcy's deck (△BDC)? Show your work. Use your calculator to round your final answer to the nearest foot. (3 points)



4. Two cars are starting from positions that are 20 miles apart. They are both headed for the same intersection, as depicted in the diagram below. Car A is traveling at 30 mph, and Car B is traveling at 45 mph. Which car will reach the intersection first?

Part I: Use the law of cosines to determine how far Car B has to travel to reach the intersection. (2 points


Part II: Use to determine the time necessary for Car A to reach the intersection. Round your answer to the nearest hundredth of an hour. (1 point)

Part III: Use to determine the time necessary for Car B to reach the intersection. (1 point)



Part IV: Which car reaches the intersection first, and by how many hours? (2 point

5. Solve for the missing length and the other two angles in the triangle below.

Part I: Use the law of cosines to find the missing third side. (2 points)

Part II: Use either the law of cosines or the law of sines to find the measure of angle C. (2 points)

Part III: Use any method you like to find the measure of angle B. (1 point)

6. Solve the triangle below.

Part I: Use the law of cosines to find the measure of angle B. (2 points)

Part II: Use the law of sines to find the measure of angle C. (2 points)
Part III: Use any method you like to find the measure of angle A. (1 point)

7. Assume two people, Swanson and Suzie, are standing 35 feet apart and are watching a boat race. At a given moment, Swanson approximates the angle formed by the lead boat, himself, and Suzie to be . Suzie approximates the angle formed by the lead boat, herself, and Swanson to be . How far is the boat from Swanson?

Part I: What is the missing angle in this triangle? (1 point)

Part II: Use the law of sines to find the distance from Swanson to the boat. (2 points)


8. Solve the following triangle for all missing sides and angles.

Part I: Find the measure of angle B. (1 point)


Part II: Use the law of sines to find the length of side a. (2 points)

Part III: Use any method to find the length of side c. (2 points)

Answer 1

You are given the mountain that has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. You are asked to find how many yards will a tourist travel from the base of the mountain to its peak. You can solve this using trigonometric function. More importantly, imagine that the mountain has a 90 - degree vertical height so that when you connect the peak of the mountain down to its base and then connected to the ski, you can form a right triangle.

Use the sine function.
sine β = opposite side / hypotenuse side
the opposite side will be the height of the mountain and the hypotenuse side will be the distance of the ski from the bottom going to the peak of the mountain

sine 40° = 100 yards / hypotenuse side
hypotenuse side = 100 yards / sine 40°
hypotenuse side = 156 yards