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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.5.50a
Chapter 4, Problem 4.5.50a

Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>




a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. 

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1
First, understand the problem setup: The dog runs from point A to point D along the beach and then swims from point D to point B in the water. We need to find the total time it takes for the dog to reach the ball at point B.
Identify the variables: Let the distance the dog runs along the beach from A to D be 'z - y', where 'z' is the total distance from A to C and 'y' is the distance from C to D. The distance the dog swims from D to B is 'x'.
Express the running time: The time taken to run from A to D is given by the formula \( \frac{z - y}{r} \), where 'r' is the running speed.
Express the swimming time: The time taken to swim from D to B is given by the formula \( \frac{x}{s} \), where 's' is the swimming speed.
Combine the times to form the total time function: The total time \( T(y) \) is the sum of the running and swimming times, which can be expressed as \( T(y) = \frac{z - y}{r} + \frac{x}{s} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this scenario, the dog’s total time to retrieve the ball is the function to be minimized. By analyzing the relationship between the distances and speeds, we can derive a function that represents the total time, which is essential for determining the optimal path the dog should take.
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Intro to Applied Optimization: Maximizing Area

Functions and Variables

A function is a mathematical relationship where each input is associated with exactly one output. In this problem, T(y) is a function of the variable y, which represents the distance the dog runs along the beach. Understanding how to express the total time as a function of y is crucial for applying calculus techniques to find the minimum time.
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Derivatives

Derivatives represent the rate of change of a function and are fundamental in finding local extrema. To minimize the total time function T(y), we will need to compute its derivative and set it to zero to find critical points. This process helps identify the optimal distance y that minimizes the time taken for the dog to reach the ball.
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Derivatives
Related Practice
Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>

a. Find the time at which the object first passes the rest position, y = 0. 

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a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

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Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.


a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Textbook Question

Sketch a graph of a function f with the following properties.


f' < 0 and f" < 0, for x < -1

Textbook Question

Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when Θ₁ = Θ₂ (see figure). <IMAGE>

Textbook Question

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>


a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.