Trigonometric Applications

In a 30-60-90 triangle, the ratio of the opposite side to the hypotenuse is 1:2. Therefore, the length of the opposite side is (1/2) * 10 cm = 5 cm.

The circumference of the Ferris wheel is π * 50 meters = 157.08 meters. Since the Ferris wheel makes one complete revolution in 10 minutes, it travels 157.08 meters in 10 minutes. Therefore, in 5 minutes, it travels half of that distance, which is 78.54 meters. Since the person boards the Ferris wheel at the bottom, they will be 78.54 meters above the ground after 5 minutes.

The ship's displacement from the port can be represented as a vector with components (100 km, 150 km). The magnitude of this vector is the distance from the port, which can be found using the Pythagorean theorem: √(100^2 + 150^2) = 250 kilometers.

The length of the cable can be found using the Pythagorean theorem. The height of the triangle formed by the cable and the two towers is 100 meters. The distance from the base of each tower to the point where the cable is secured to the ground is 50 meters. Therefore, the length of the cable is √(100^2 + 50^2) = 175 meters.

The ladder, the wall, and the ground form a right triangle. The length of the ladder is the hypotenuse of this triangle. The distance from the base of the ladder to the wall is the adjacent side. The height of the ladder up the wall is the opposite side. Using the Pythagorean theorem, we can find the height of the ladder: √(10^2 - 6^2) = 8 feet.

The angle of elevation, the height of the kite, and the distance from the point on the ground to the base of the kite form a right triangle. The height of the kite is the opposite side of this triangle. The distance from the point on the ground to the base of the kite is the adjacent side. Using the tangent function, we can find the distance from the point on the ground to the base of the kite: tan(30°) = 100 meters / x, where x is the distance from the point on the ground to the base of the kite. Solving for x, we get x = 173.21 meters.

The angle of elevation, the height of the building, and the distance from the surveyor to the base of the building form a right triangle. The height of the building is the opposite side of this triangle. The distance from the surveyor to the base of the building is the adjacent side. Using the tangent function, we can find the height of the building: tan(45°) = h / 100 meters, where h is the height of the building. Solving for h, we get h = 141.42 meters.

The boat's displacement from the dock can be represented as a vector with components (20 km, 30 km). The magnitude of this vector is the distance from the dock, which can be found using the Pythagorean theorem: √(20^2 + 30^2) = 36.06 kilometers.

The circumference of the Ferris wheel is π * 40 meters = 125.66 meters. Since the Ferris wheel makes one complete revolution in 8 minutes, it travels 125.66 meters in 8 minutes. Therefore, in 4 minutes, it travels half of that distance, which is 62.83 meters. Since the person boards the Ferris wheel at the bottom, they will be 62.83 meters above the ground after 4 minutes.

The ladder, the wall, and the ground form a right triangle. The length of the ladder is the hypotenuse of this triangle. The distance from the base of the ladder to the wall is the adjacent side. The height of the ladder up the wall is the opposite side. Using the Pythagorean theorem, we can find the height of the ladder: √(12^2 - 4^2) = 10 feet.

The angle of elevation, the height of the kite, and the distance from the point on the ground to the base of the kite form a right triangle. The height of the kite is the opposite side of this triangle. The distance from the point on the ground to the base of the kite is the adjacent side. Using the tangent function, we can find the distance from the point on the ground to the base of the kite: tan(60°) = 50 meters / x, where x is the distance from the point on the ground to the base of the kite. Solving for x, we get x = 43.30 meters.

The angle of elevation, the height of the building, and the distance from the surveyor to the base of the building form a right triangle. The height of the building is the opposite side of this triangle. The distance from the surveyor to the base of the building is the adjacent side. Using the tangent function, we can find the height of the building: tan(30°) = h / 50 meters, where h is the height of the building. Solving for h, we get h = 25 meters.

The boat's displacement from the dock can be represented as a vector with components (10 km, 20 km). The magnitude of this vector is the distance from the dock, which can be found using the Pythagorean theorem: √(10^2 + 20^2) = 22.36 kilometers.

The circumference of the Ferris wheel is π * 30 meters = 94.25 meters. Since the Ferris wheel makes one complete revolution in 6 minutes, it travels 94.25 meters in 6 minutes. Therefore, in 3 minutes, it travels half of that distance, which is 47.13 meters. Since the person boards the Ferris wheel at the bottom, they will be 47.13 meters above the ground after 3 minutes.

The ladder, the wall, and the ground form a right triangle. The length of the ladder is the hypotenuse of this triangle. The distance from the base of the ladder to the wall is the adjacent side. The height of the ladder up the wall is the opposite side. Using the Pythagorean theorem, we can find the height of the ladder: √(8^2 - 3^2) = 5 feet.