Final Answer:
a. The diagram shows a vertical cliff with a height of 450 m, a ship located x meters from the base, and the angle of elevation to the top of the cliff marked as 23°.
b. To calculate the value of x, we can use the tangent of the angle of elevation. Tangent is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In this case, tan(23°) = 450 / x. Solving for x, we get x ≈ 1132.69 m.
Step-by-step explanation:
a. The diagram illustrates a right-angled triangle formed by the vertical cliff, the horizontal distance from the ship to the base of the cliff (x), and the line of sight from the ship to the top of the cliff. The given angle of elevation (23°) is the angle between the line of sight and the horizontal ground. This visualization helps establish the components of the problem.
b. Applying the tangent function, tan(θ) = opposite / adjacent, where θ is the angle of elevation, we have tan(23°) = 450 / x. Solving for x, we rearrange the equation to x = 450 / tan(23°). Utilizing a calculator, we find x ≈ 1132.69 m. This value represents the horizontal distance from the ship to the base of the cliff.
Understanding trigonometric functions, especially in the context of angles of elevation or depression, is crucial in solving real-world problems involving heights and distances. In this scenario, the tangent function provides a straightforward approach to determine the horizontal distance from the ship to the base of the cliff given the height and angle of elevation.