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AP Calculus AB: 7.3.2 The Fence Problem

Mathematics8 CardsCreated 3 months ago

This flashcard set focuses on solving optimization problems involving fences, where the goal is to maximize the enclosed area using a fixed length of fencing. It explains the process of relating perimeter and area, setting up and differentiating the area function, and finding critical points to determine maximum values, with practical examples like garden and chicken pen problems.

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The Fence Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • The fence problem involves maximizing the fenced-in area without changing the amount of fence used. Set the derivative of the area function equal to 0 and solve.

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Key Terms

Term
Definition

The Fence Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • ...

note

  • There are actually two types of maxima and minima. A
    relative maximum occurs anytime you have a summit, but it does not have to be that ...

A man decides to create a rectangular garden along the back side of his home and wants to build a fence to enclose it. As luck would have it, he happens to have 60 feet of fencing materials laying around in his garage.

Keeping in mind that one side of the garden would be flush with the house, what is the maximum area that the man could enclose with his fence?

A = 450

A farmer is building a chicken pen in the shape of a rectangle and has 100 feet of fencing material. If a chicken needed 2 square feet to live, what would be the maximum number of chickens the farmer could keep alive in the pen?

312 chickens

A couple of newlyweds decides to surround their house with a brand new white picket fence. They purchase 120 feet of fencing materials. If they build the fence in the shape of the rectangle that will maximize the area enclosed by the fence, what is the area that the fence will enclose?

900 ft 2

A farmer needs a new pen for some of his livestock and decides to build a rectangular fence. The farmer has 60 feet of fence. What is the greatest area the farmer could enclose?

225 feet 2

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AP Calculus AB: 7.3.2 The Fence Problem
TermDefinition

The Fence Problem

  • Values that make the derivative of a function equal to 0 or undefined are candidates for maxima and minima of the function.

  • The fence problem involves maximizing the fenced-in area without changing the amount of fence used. Set the derivative of the area function equal to 0 and solve.

note

  • There are actually two types of maxima and minima. A
    relative maximum occurs anytime you have a summit, but it does not have to be that absolute highest point the graph attains. The absolute maximum is the highest the graph can get.

  • Likewise, a relative minimum occurs anytime you have a valley. The absolute minimum is the lowest the graph can get.

  • When solving a problem, think about what you want, figure out what you know, and relate the two.

  • The question asks you to find the dimensions of the fence that would enclose the greatest area, using the farmhouse as one side of the pen. So the question asks for you to maximize the area.

  • You are told that you have 100 feet of fencing material. You also know that the material makes up three sides of the perimeter of the rectangle.

  • When you solve the perimeter equation for x, you can then substitute that expression into the area equation. This relates what you want and what you know. Once you have the area equation in terms of a single variable, you can differentiate.

  • Set the derivative of the area equation equal to 0 to find potential maximum points.

  • Be careful which variable you use when you solve this
    problem. Do not mix them up!

A man decides to create a rectangular garden along the back side of his home and wants to build a fence to enclose it. As luck would have it, he happens to have 60 feet of fencing materials laying around in his garage.

Keeping in mind that one side of the garden would be flush with the house, what is the maximum area that the man could enclose with his fence?

A = 450

A farmer is building a chicken pen in the shape of a rectangle and has 100 feet of fencing material. If a chicken needed 2 square feet to live, what would be the maximum number of chickens the farmer could keep alive in the pen?

312 chickens

A couple of newlyweds decides to surround their house with a brand new white picket fence. They purchase 120 feet of fencing materials. If they build the fence in the shape of the rectangle that will maximize the area enclosed by the fence, what is the area that the fence will enclose?

900 ft 2

A farmer needs a new pen for some of his livestock and decides to build a rectangular fence. The farmer has 60 feet of fence. What is the greatest area the farmer could enclose?

225 feet 2

An expecting mother wants to convert a corner of the nursery in her house into a rectangular playpen by enclosing the corner of the room with two fences. If the woman has 10 feet of fencing material, what is the maximum area the playpen could encompass?

25 square feet

An interstellar colonist is raising Snuffles and decides to build a new Snuffle pen along the outside of his rectangular barn. Snuffles like to live in right triangles, so the colonist wants the pen to be in the shape of a right triangle, with one side flush with the barn. The colonist has 60 feet of fence material. Using the barn as one side of the right triangle, what is the maximum area the colonist can enclose?

450 ft2