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A power line is to be constructed from a power station at point A to an island at point C, which is 4 mi directly out in the wate Point B is 4 mi downshore from the power station at A. It costs $3600 per mile to lay the power line under water and $2000 ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could v length of CS is √16+x².)

A power line is to be constructed from a power station at point A to an island at point C, which is 4 mi directly out in the wate Point B is 4 mi downshore from the power station at A. It costs $3600 per mile to lay the power line under water and $2000 ground. At what point S downshore from A should the line come to the shore in order to minimize cost? Note that S could v length of CS is √16+x².)

Calculus: Early Transcendentals
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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### Problem Description A power line is planned to be constructed from a power station at point A to an island at point C, which is 4 miles directly out in the water from point B on the shore. Point B is located 4 miles downshore from the power station at A. The cost to lay the power line is $3600 per mile underwater and $2000 per mile underground. We need to determine at what point, S, downshore from A, the line should come to shore in order to minimize the cost of construction. Note that point S could potentially be point B or A. To assist with visualizing the problem, a diagram is provided: ### Diagram Explanation The diagram shows: 1. Point A representing the power station on the shore. 2. Point B, which is 4 miles downshore from point A. 3. Point C, located 4 miles directly out into the water from point B. 4. Point S is denoted somewhere along the shore between A and B. 5. The length of CS, which can be expressed as \(\sqrt{16 + x^2}\), where \(x\) is the distance from B to S along the shore. ### Mathematical Representation - Let \(x\) be the distance from B to the point S along the shore. - Therefore, the distance from A to S is \(4 - x\). - The length of the underwater line segment CS can be represented as \(\sqrt{16 + x^2}\). ### Objective Determine the value of \(x\) that minimizes the total cost of construction, taking into consideration the costs per mile for underwater and underground power lines. ### Solution Format S is \( \boxed{\ \ \ \ \ \ \ \ \ \ } \) miles from A. (Round to two decimal places as needed.) --- ### Step-by-Step Solution (for further implementation on the educational website) 1. Define the cost function for constructing the power line. 2. Use calculus (specifically optimization techniques) to find the value of \(x\) that minimizes the total cost. 3. Verify the solution to ensure it results in the minimal cost compared to possible boundary points (A and B). This problem integrates concepts from geometry, calculus, and economics, offering a practical application scenario for optimization methods.
Transcribed Image Text:### Problem Description A power line is planned to be constructed from a power station at point A to an island at point C, which is 4 miles directly out in the water from point B on the shore. Point B is located 4 miles downshore from the power station at A. The cost to lay the power line is $3600 per mile underwater and $2000 per mile underground. We need to determine at what point, S, downshore from A, the line should come to shore in order to minimize the cost of construction. Note that point S could potentially be point B or A. To assist with visualizing the problem, a diagram is provided: ### Diagram Explanation The diagram shows: 1. Point A representing the power station on the shore. 2. Point B, which is 4 miles downshore from point A. 3. Point C, located 4 miles directly out into the water from point B. 4. Point S is denoted somewhere along the shore between A and B. 5. The length of CS, which can be expressed as \(\sqrt{16 + x^2}\), where \(x\) is the distance from B to S along the shore. ### Mathematical Representation - Let \(x\) be the distance from B to the point S along the shore. - Therefore, the distance from A to S is \(4 - x\). - The length of the underwater line segment CS can be represented as \(\sqrt{16 + x^2}\). ### Objective Determine the value of \(x\) that minimizes the total cost of construction, taking into consideration the costs per mile for underwater and underground power lines. ### Solution Format S is \( \boxed{\ \ \ \ \ \ \ \ \ \ } \) miles from A. (Round to two decimal places as needed.) --- ### Step-by-Step Solution (for further implementation on the educational website) 1. Define the cost function for constructing the power line. 2. Use calculus (specifically optimization techniques) to find the value of \(x\) that minimizes the total cost. 3. Verify the solution to ensure it results in the minimal cost compared to possible boundary points (A and B). This problem integrates concepts from geometry, calculus, and economics, offering a practical application scenario for optimization methods.
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