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AP Calculus AB: 7.3.5 The Wire-Cutting Problem

Mathematics5 CardsCreated 3 months ago

These flashcards explore optimization problems involving cutting a wire into segments to form geometric shapes, such as squares, circles, and polygons, aiming to minimize the total enclosed area. They cover expressing areas in terms of one variable, differentiating to find minimum values, and applying geometric formulas to real-world contexts.

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note

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

  • The wire cutting problem involves minimizing the sum of the areas of a square and a circle formed from a fixed length of material.

  • For a given perimeter, a circle encloses more area than a square does.

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Key Terms

Term
Definition

note

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

  • ...

note

  • In this problem you are asked to find where a wire
    should be cut so that the area made by the resulting
    two pieces is minimized.

    <...

A 16 inch wire is cut in two and shaped into two squares. What is the minimum possible sum of the two areas?

Asum = 8 in^2

A 12 inch wire is being cut into two pieces which are then shaped into a circle and a regular octagon of maximum area. What is the sum of the perimeters?

12 inches

A 24 inch wire is cut in two and shaped into a square and a regular octagon. What is the minimum possible sum of the two areas?AOctagon=(2+2√2)s^2

Asum  = 19.689in^2

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AP Calculus AB: 7.3.5 The Wire-Cutting Problem
TermDefinition

note

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

  • The wire cutting problem involves minimizing the sum of the areas of a square and a circle formed from a fixed length of material.

  • For a given perimeter, a circle encloses more area than a square does.

note

  • In this problem you are asked to find where a wire
    should be cut so that the area made by the resulting
    two pieces is minimized.

  • You know the length of the wire, the formulas for the
    area and perimeter of the square, and the formulas
    for the area and circumference of the circle.

  • Since the perimeter plus the circumference equals
    the total length of the wire, you can relate the area
    to the length of the wire by expressing the perimeter
    (or sides of the square) in terms of the
    circumference. This will enable you to express the
    area in terms of a single variable, namely c.

  • Differentiate the area to find the minimum
    candidate. Do not forget to check that the candidate
    is indeed a minimum.

  • Knowing the circumference tells you where to cut
    the wire.

  • Circles pack area more efficiently than squares. So
    if you want to minimize the sum of the two areas, it
    makes sense that you would want the square to be
    bigger.

A 16 inch wire is cut in two and shaped into two squares. What is the minimum possible sum of the two areas?

Asum = 8 in^2

A 12 inch wire is being cut into two pieces which are then shaped into a circle and a regular octagon of maximum area. What is the sum of the perimeters?

12 inches

A 24 inch wire is cut in two and shaped into a square and a regular octagon. What is the minimum possible sum of the two areas?AOctagon=(2+2√2)s^2

Asum  = 19.689in^2