Textbook Question
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.11.49
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?
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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6:04Introduction to Trigonometric Functions
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.
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06:29Derivatives Applied To Velocity
Related Practice
Textbook Question
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Find the derivative of the following functions.
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Find the derivative of the following functions.
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Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)
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