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Mathematics /AP Calculus AB: 7.3.5 The Wire-Cutting Problem

AP Calculus AB: 7.3.5 The Wire-Cutting Problem

Mathematics5 CardsCreated 7 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

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. 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