Trigonometry Real-World Applications 1 — Questions and Answers
Question 1: A surveyor measures the angle of elevation to the top of a building as 45∘. If the surveyor is standing 50 meters away from the building, what is the building’s height?
- 25 meters
- 50 meters (Correct answer)
- 75 meters
- 100 meters
Correct answer: 50 meters
This problem involves a right-angled triangle where the angle of elevation is 45 degrees. In a right triangle, if one acute angle is 45 degrees, the other acute angle must also be 45 degrees, making it an isosceles right triangle. This means the side opposite the 45-degree angle (height) is equal to the side adjacent to it (distance from the building), so the height is 50 meters.
Question 2: A ship travels 20 km due north, then turns and travels 15 km due east. What is the direct distance between the ship’s starting and ending points?
- 25 km (Correct answer)
- 20 km
- 15 km
- 10 km
Correct answer: 25 km
The ship's journey forms a right-angled triangle, with the 20 km north and 15 km east movements as the two perpendicular sides. The direct distance from the starting to ending point is the hypotenuse of this triangle. Using the Pythagorean theorem (a² + b² = c²), 20² + 15² = 400 + 225 = 625, and the square root of 625 is 25 km.
Question 3: The height of a flagpole casts a shadow 12 meters long when the sun’s elevation angle is 30∘. What is the height of the flagpole?
- 6 meters
- 6 √3meters (Correct answer)
- 12 meters
- 12 √3 meters
Correct answer: 6 √3meters
This problem forms a right-angled triangle where the flagpole's height is the opposite side, and the shadow length is the adjacent side to the angle of elevation. Using the tangent function, tan(angle) = opposite/adjacent, so tan(30°) = height / 12. Since tan(30°) = 1/√3, the height = 12 * (1/√3) = 12√3 / 3 = 4√3 meters. (Note: There is a discrepancy between the calculated answer 4√3 and the provided correct answer 6√3. If the correct answer is 6√3, the shadow length would need to be 18 meters, as 18 * tan(30°) = 18 * (1/√3) = 6√3.)
Question 4: An engineer needs to calculate the height of a tower using trigonometry. From a distance of 100 meters, the angle of elevation to the top of the tower is 60∘. What is the height of the tower?
- 50 meters
- 100 meters
- 100 √3meters (Correct answer)
- 50 √3meters
Correct answer: 100 √3meters
This problem involves a right-angled triangle where the height of the tower is the opposite side, and the distance from the tower is the adjacent side to the angle of elevation. Using the tangent function, tan(angle) = opposite/adjacent, so tan(60°) = height / 100. Since tan(60°) = √3, the height of the tower is 100 * √3 meters.
Question 5: A ladder leans against a wall, forming a 60∘ angle with the ground. If the ladder is10 feet long, how high up the wall does it reach?
- 5 feet
- 5 √3 feet (Correct answer)
- 10 feet
- 10 √3 feet
Correct answer: 5 √3 feet
This scenario forms a right-angled triangle where the ladder is the hypotenuse, and the height it reaches up the wall is the side opposite the 60-degree angle with the ground. Using the sine function, sin(angle) = opposite/hypotenuse, so sin(60°) = height / 10. Since sin(60°) = √3/2, the height = 10 * (√3/2) = 5√3 feet.
Trigonometry Real-World Applications 1
A surveyor measures the angle of elevation to the top of a building as 45∘.
If the surveyor is standing 50 meters away from the building, what is the building’s height?