Cassidy is a lifeguard and spots a drowning child 60 meters along the shore and 50 meters from the shore to the child. Cassidy runs along the shore for a while and then jumps into the water and swims from there directly to the child. Cassidy can run at a rate of 3 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should Cassidy run before jumping into the water in order to save the child? Round your answer to three decimal places.
Cassidy is a lifeguard and spots a drowning child 60 meters along the shore and 50 meters from the shore to the child. Cassidy runs along the shore for a while and then jumps into the water and swims from there directly to the child. Cassidy can run at a rate of 3 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should Cassidy run before jumping into the water in order to save the child? Round your answer to three decimal places.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Lifeguard Rescue Scenario Calculation**
**Problem Statement:**
Cassidy is a lifeguard and spots a drowning child 60 meters along the shore and 50 meters from the shore to the child. Cassidy runs along the shore for a while and then jumps into the water and swims from there directly to the child. Cassidy can run at a rate of 3 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should Cassidy run before jumping into the water in order to save the child? Round your answer to three decimal places.
**Explanation:**
*Graph Description:*
The graph provided is a standard Cartesian coordinate system where the x-axis represents the distance along the shore in meters, and the y-axis represents the distance from the shore in meters. The 60 meters along the shore and 50 meters from the shore to the child are marked on the graph, indicating the child’s location relative to Cassidy's starting point at the origin (0,0).
*Key Points and Given Data:*
- Distance along the shore to the child: 60 meters
- Distance from the shore to the child: 50 meters
- Running speed of Cassidy: 3 meters per second
- Swimming speed of Cassidy: 0.8 meters per second
Cassidy’s total time to reach the child is the sum of the time running and the time swimming. Let \( x \) be the distance Cassidy runs along the shore before jumping into the water.
*Objective:*
To determine the optimal point \( x \) such that the total rescue time is minimized.
*Steps to Solve:*
1. Calculate the running time along the shore for distance \( x \).
2. Calculate the remaining distance Cassidy swims from the shore to the child.
3. Use the Pythagorean theorem to determine the swimming distance from point \( x \) on the shore directly to the child.
4. Calculate the swimming time for the remaining distance.
5. Add the running and swimming time to get the total time taken.
6. Minimize this total time with respect to \( x \).
By solving the above, the distance \( x \) along the shore is determined, ensuring the fastest rescue path.
**Solution:**
To be solved using appropriate mathematical techniques (like calculus or numerical methods) to find the exact distance \( x \).
\[ \text{Answer (rounded to three decimal places):} \ ________ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F35c309d3-406a-498d-9918-e953e5869332%2F5k4kfsr_processed.png&w=3840&q=75)
Transcribed Image Text:**Lifeguard Rescue Scenario Calculation**
**Problem Statement:**
Cassidy is a lifeguard and spots a drowning child 60 meters along the shore and 50 meters from the shore to the child. Cassidy runs along the shore for a while and then jumps into the water and swims from there directly to the child. Cassidy can run at a rate of 3 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should Cassidy run before jumping into the water in order to save the child? Round your answer to three decimal places.
**Explanation:**
*Graph Description:*
The graph provided is a standard Cartesian coordinate system where the x-axis represents the distance along the shore in meters, and the y-axis represents the distance from the shore in meters. The 60 meters along the shore and 50 meters from the shore to the child are marked on the graph, indicating the child’s location relative to Cassidy's starting point at the origin (0,0).
*Key Points and Given Data:*
- Distance along the shore to the child: 60 meters
- Distance from the shore to the child: 50 meters
- Running speed of Cassidy: 3 meters per second
- Swimming speed of Cassidy: 0.8 meters per second
Cassidy’s total time to reach the child is the sum of the time running and the time swimming. Let \( x \) be the distance Cassidy runs along the shore before jumping into the water.
*Objective:*
To determine the optimal point \( x \) such that the total rescue time is minimized.
*Steps to Solve:*
1. Calculate the running time along the shore for distance \( x \).
2. Calculate the remaining distance Cassidy swims from the shore to the child.
3. Use the Pythagorean theorem to determine the swimming distance from point \( x \) on the shore directly to the child.
4. Calculate the swimming time for the remaining distance.
5. Add the running and swimming time to get the total time taken.
6. Minimize this total time with respect to \( x \).
By solving the above, the distance \( x \) along the shore is determined, ensuring the fastest rescue path.
**Solution:**
To be solved using appropriate mathematical techniques (like calculus or numerical methods) to find the exact distance \( x \).
\[ \text{Answer (rounded to three decimal places):} \ ________ \]
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