Textbook Question
The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot.
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 68b
Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .
b. When do the particles collide?
Verified step by step guidance1
To find when the particles collide, we need to determine when their positions are equal. Set the position functions equal to each other: \( \cos t = \cos (t + \frac{\pi}{4}) \).
Recall the trigonometric identity for cosine: \( \cos A = \cos B \) implies \( A = 2n\pi \pm B \) for integer \( n \). Apply this identity to the equation \( \cos t = \cos (t + \frac{\pi}{4}) \).
This gives us two possible equations: \( t = 2n\pi + (t + \frac{\pi}{4}) \) and \( t = 2n\pi - (t + \frac{\pi}{4}) \).
Solve each equation for \( t \). For the first equation, simplify to find \( 0 = 2n\pi + \frac{\pi}{4} \), which is not possible. For the second equation, simplify to find \( 2t = 2n\pi - \frac{\pi}{4} \).
Solve the second equation for \( t \) to find the specific times when the particles collide. This will give you the values of \( t \) in terms of \( n \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Particle Position Functions
In this context, the positions of the two particles are given by the functions s₁ = cos(t) and s₂ = cos(t + π/4). These functions describe how the position of each particle changes over time, with 't' representing time. Understanding these functions is crucial for determining when the particles are at the same position, which is essential for solving the collision problem.
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Collision Condition
For two particles to collide, their positions must be equal at the same time. This means we need to set the position functions equal to each other: cos(t) = cos(t + π/4). Solving this equation will yield the values of 't' at which the particles occupy the same position on the s-axis, indicating a collision.
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Trigonometric Identities
Trigonometric identities, such as the cosine addition formula, are essential for simplifying and solving equations involving trigonometric functions. In this case, using the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b) can help in transforming the collision condition into a more manageable form, allowing for the determination of the specific times 't' when the particles collide.
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Related Practice
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Textbook Question
Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .
a. What is the farthest apart the particles ever get?
Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local maxima at (1, 1) and (3, 3)