A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

검증된 단계별 안내

First, identify the components of the submarine's velocity. The submarine is diving at an angle of 20° below the horizontal, which means its velocity can be split into horizontal and vertical components.

The horizontal component of the submarine's velocity is the same as the ship's velocity, which is 10 km/hr. This is because the submarine maintains a position directly below the ship.

To find the vertical component of the submarine's velocity, use trigonometry. The vertical component can be found using the sine function: \( v_{vertical} = v_{total} \cdot \sin(20°) \).

Since the horizontal component is 10 km/hr, use the cosine function to find the total velocity of the submarine: \( v_{total} = \frac{10}{\cos(20°)} \).

Substitute the total velocity into the equation for the vertical component: \( v_{vertical} = \frac{10}{\cos(20°)} \cdot \sin(20°) \). This will give you the rate at which the submarine's altitude is decreasing.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to relate the horizontal speed of the ship to the vertical speed of the submarine as it dives. By using derivatives, we can establish a relationship between the rates of change of the ship's position and the submarine's altitude.

Trigonometric Functions

Trigonometric functions, particularly sine and cosine, are essential for analyzing angles and distances in this scenario. The angle of 20° below the horizontal allows us to use these functions to determine the vertical component of the submarine's movement. Understanding how to decompose the submarine's velocity into horizontal and vertical components is crucial for solving the problem.

Velocity Components

Velocity components refer to breaking down a velocity vector into its horizontal and vertical parts. In this case, the submarine's velocity can be split into a horizontal component (which matches the ship's speed) and a vertical component (which represents the rate of altitude decrease). This decomposition is vital for applying the related rates concept effectively.