Summer Adventure in New York City: Trigonometry Challenges

As the students navigate through Central Park, they come across a sculpture that is 12 feet tall. From a distance of 10 feet from the base of the sculpture, one student measures the angle of elevation to the top of the sculpture. What is the angle of elevation they recorded? (Use tan θ = opposite/adjacent.)

Question: What is the angle of elevation to the top of the sculpture?

Answer: The angle of elevation is approximately 48.37°.

While exploring Times Square, the group decides to form a triangle with their positions. One student is located 40 meters south of another, who is located 30 meters east of a third student. What is the angle at the third student’s position with respect to the other two students? (Hint: Use the sine or cosine rule.)

Question: What is the angle at the third student's position?

Answer: The angle is approximately 36.87°.

At the top of the Empire State Building, the students notice that the distance from the observation deck to a specific point on the ground is 200 meters, with a 45° angle of depression to that point. How tall is the Empire State Building from the ground?

Question: What is the building's height?

Answer: The building's height is 200 meters.

Determined to find the best viewpoint, the students calculate the height of a building across the street. They measure a 30° angle of elevation from their position 25 feet away from the base of the building. How tall is the building? (Use tan θ for calculations.)

Question: What is the height of the building?

Answer: The height of the building is approximately 14.5 feet.

While waiting for their subway train, the group sees a triangular fountain where the base measures 8 feet and the height is 6 feet. What is the angle of elevation from one end of the base to the top of the fountain? (Use tan θ = opposite/adjacent.)

Question: What is the angle of elevation to the top of the fountain?

Answer: The angle of elevation is approximately 36.87°.

During their visit to the Brooklyn Bridge, the students stand at a point where they can see the bridge spanning a distance of 250 meters away. They measure that the angle of elevation to the highest point of the bridge is 25°. How high is the bridge at that point? (Use the formula: height = distance * tan(angle).)

Question: How high is the bridge at that point?

Answer: The height of the bridge is approximately 11.66 meters.

The students decide to measure the angle of elevation to the top of the Flatiron Building. From a distance of 30 meters on the opposite sidewalk, they find the height of the building to be 8 meters when they sit on a bench. What is the angle of elevation? (Use tan θ = opposite/adjacent.)

Question: What is the angle of elevation to the top of the Flatiron Building?

Answer: The angle of elevation is approximately 14.74°.

While exploring the Bronx Zoo, the students notice that a giraffe is directly 40 meters from where they are standing, and they observe it standing at an angle of 60° from their line of sight. How tall is the giraffe? (Use the sine function.)

Question: How tall is the giraffe?

Answer: The giraffe's height is approximately 34.64 meters.

As the students prepare to take the ferry to Staten Island, they notice that the ferry is docked at a 10° angle relative to the pier. If the distance from the dock to the ferry is 50 meters, how far away is the ferry from the pier? (Use sin θ = opposite/hypotenuse.)

Question: What is the distance from the ferry to the pier?

Answer: The distance from the ferry to the pier is approximately 8.76 meters.

Finally, as they finish their adventure, the students calculate the angle at which they need to look up to see the Statue of Liberty from 100 meters away. If the statue's height is 93 meters, what is the angle of elevation? (Use the formula: angle = tan⁻¹(height/distance).)

Question: What is the angle of elevation to the Statue of Liberty?

Answer: The angle of elevation is approximately 45.57°.