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A piece of wire 8 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? 8 m (b) How much wire should be used for the square in order to minimize the total area?

A piece of wire 8 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? 8 m (b) How much wire should be used for the square in order to minimize the total area?

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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A piece of wire 8 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? 8 m (b) How much wire should be used for the square in order to minimize the total area? m Enhanced Feedback Please try again and draw a diagram. Keep in mind that the area of a square with edge a is As = a? and the area of an equilateral triangle with edge b is A =, Let x be the perimeter of the square, which means x = 4a, and y be the 4 perimeter of the triangle, which means y = 3b. Find a relationship between x and y, considering that the wire's length / is a constant and 1 = x + y. Rewrite the total area A = Ag + A, as a function of one variable. Use calculus to find the edges of the square and the triangle that maximize the area; then find the edges that minimize the area.
Transcribed Image Text:A piece of wire 8 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximize the total area? 8 m (b) How much wire should be used for the square in order to minimize the total area? m Enhanced Feedback Please try again and draw a diagram. Keep in mind that the area of a square with edge a is As = a? and the area of an equilateral triangle with edge b is A =, Let x be the perimeter of the square, which means x = 4a, and y be the 4 perimeter of the triangle, which means y = 3b. Find a relationship between x and y, considering that the wire's length / is a constant and 1 = x + y. Rewrite the total area A = Ag + A, as a function of one variable. Use calculus to find the edges of the square and the triangle that maximize the area; then find the edges that minimize the area.
Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of each of two sides and three dividers. Let y denote the length of the other two sides. (c) Write an expression for the total area A in terms of both x and y. A = xy (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. 750x – 5x2 A(x) = (f) Finish solving the problem by finding the largest area. 14062.5 ft2
Transcribed Image Text:Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of each of two sides and three dividers. Let y denote the length of the other two sides. (c) Write an expression for the total area A in terms of both x and y. A = xy (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. 750x – 5x2 A(x) = (f) Finish solving the problem by finding the largest area. 14062.5 ft2
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