Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 4 - Applications of the Derivative문제 4.5.50a
4장, 문제 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. 

검증된 단계별 안내
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 \( T(y) = \frac{z - y}{r} + \frac{x}{s} \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m

주요 개념

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

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.
추천 영상:
10:13
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.
추천 영상:
6:13
Exponential Functions

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.
추천 영상:
가이드 코스
05:44
Derivatives
관련 실천
교과서 질문

Optimal soda can


a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.

292
views
교과서 질문

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The function f(x) = √x has a local maximum on the interval [0,∞).

234
views
교과서 질문

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


f' < 0 and f" < 0, for x < 3

347
views
교과서 질문

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


a. Plot a graph of the curve when a = 3.

997
views
교과서 질문

{Use of Tech} Investment problem A one-time investment of \$2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is a(r) = 2500(1 + r/12)⁶⁰. 


a. Use Newton’s method to find the value of r if the goal is to have \$3200 in the account after 5 years.

173
views
교과서 질문

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51). 


a. Without using a calculator, find the values of a, with 0 ≤ a ≤ 4, such that f  has a fixed point. Give the fixed point in terms of a. 

247
views