**Problem 15: Optimal Fencing for a Rectangular Enclosure**A farmer wishes to enclose a rectangular region using 500 feet of fencing. One side of this region will be alongside a barn, which does not require any fencing. Let $ x $ represent the length of one of the parallel sides perpendicular to the barn.**Tasks:**a) **Express the length of the remaining side in terms of $ x $.**b) **Determine a function $ A $ that represents the area of the region in terms of $ x $.**c) **Calculate the value of $ x $ that maximizes the area, and find the maximum area.****Diagram:**- The illustration accompanying the problem shows a rectangular area with the barn wall labeled as one side.- The two sides perpendicular to the barn are labeled $ x $.- The side opposite the barn, which needs fencing, is labeled. Since the total amount of fencing used is 500 feet and two sides are $ x $, the expression for the remaining side’s length can be deduced.**Solution Outline:**- **a)** The length of the opposite side to the barn can be expressed as $ 500 - 2x $.- **b)** The area $ A $ is given by the formula for the area of a rectangle: $ A = x \times (500 - 2x) $.- **c)** To find the maximum area, derive the area function $ A(x) $, set the derivative to zero, and solve for $ x $. Calculate the maximum area using this optimal $ x $.

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Description: **Problem 15: Optimal Fencing for a Rectangular Enclosure**A farmer wishes to enclose a rectangular region using 500 feet of fencing. One side of this region will be alongside a barn, which does not require any fencing. Let $ x $ represent the len

(a) The length of the fencing is 500 feet that is the perimeter of the region. Consider the perimeter of the region is 500 feet and the length of the one side is x. Consider the length of the other side is y then, 2x+y=500x+y=5002y=250-x Thus, the length of the other side is 250-x.(b) Consider the area of the rectangle is, A=LB Then, for the given region. A=x250-x=250x-x2 The expression for the area of the region in terms of x as, A=250x-x2(c) The value of x to obtain the maximum area. A=250x-x2dAdx=ddx250x-x20=250-2x2x=250x=2502x=125 Thus, the required value of x is 125.

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Algebra

15. A farmer wishes to enclose a rectangular region. He has 500 feet of fencing and plans to use his barn as one side of the enclosure. Let x represent the length of one of parallel sides of the fencing. a) Express the length of the remaining side in terms of x. b) Determine a function A that represents the area of the region in terms of x. 500x c) What is the value of x that will give a maximum area and what will this area be?

15. A farmer wishes to enclose a rectangular region. He has 500 feet of fencing and plans to use his barn as one side of the enclosure. Let x represent the length of one of parallel sides of the fencing. a) Express the length of the remaining side in terms of x. b) Determine a function A that represents the area of the region in terms of x. 500x c) What is the value of x that will give a maximum area and what will this area be?

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Problem 15: Optimal Fencing for a Rectangular Enclosure**

A farmer wishes to enclose a rectangular region using 500 feet of fencing. One side of this region will be alongside a barn, which does not require any fencing. Let $ x $ represent the length of one of the parallel sides perpendicular to the barn.

**Tasks:**

a) **Express the length of the remaining side in terms of $ x $.**

b) **Determine a function $ A $ that represents the area of the region in terms of $ x $.**

c) **Calculate the value of $ x $ that maximizes the area, and find the maximum area.**

**Diagram:**

- The illustration accompanying the problem shows a rectangular area with the barn wall labeled as one side.
- The two sides perpendicular to the barn are labeled $ x $.
- The side opposite the barn, which needs fencing, is labeled. Since the total amount of fencing used is 500 feet and two sides are $ x $, the expression for the remaining side’s length can be deduced.

**Solution Outline:**

- **a)** The length of the opposite side to the barn can be expressed as $ 500 - 2x $.
- **b)** The area $ A $ is given by the formula for the area of a rectangle: $ A = x \times (500 - 2x) $.
- **c)** To find the maximum area, derive the area function $ A(x) $, set the derivative to zero, and solve for $ x $. Calculate the maximum area using this optimal $ x $.

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