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AP Calculus AB: 7.3.3 The Box Problem

Mathematics8 CardsCreated 3 months ago

This set of flashcards covers optimization problems involving constructing an open box from a rectangular sheet by cutting squares from each corner. It explains how to formulate the volume function in one variable, find critical points by differentiation, and interpret results to maximize volume, illustrated with practical examples involving material waste and box dimensions.

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The Box Problem

The box problem involves maximizing the volume of an open box constructed from a given rectangular sheet of material.

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Key Terms

Term
Definition

The Box Problem

The box problem involves maximizing the volume of an open box constructed from a given rectangular sheet of material.

note

  • In the box problem, you are asked to maximize the volume of a box constructed from a rectangular sheet.

  • You know the dimensi...

A man is constructing a box out of a 20 × 16 sheet of metal. He opts to cut a 3 × 3 square off of each corner of the sheet and then fold the sides upwards. What is the area of the wasted material? Dimensions are in inches.

36 square inches

A particular company manufactures cardboard boxes of many different volumes. They always start with a flat sheet of cardboard. By cutting an equal sized square from each corner and folding the sides upwards, they make a box. If the cardboard sheet is 18 × 16 inches and you cut a 2 × 2 square from each corner, what would be the volume of the resulting box?

336 cubic inches

What is the greatest amount of sand you could put in a sandbox made from a 4 foot by 6 foot piece of plywood? Remember, the sandbox will be made by cutting equal sized squares from the corners and then “folding” the sides upwards.

8.45 cubic feet

What is the maximum volume of an enclosed rectangular box with surface area equal to 54 sq. ft and a height of 3 ft?

V = 27 cubic feet

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AP Calculus AB: 7.3.3 The Box Problem
TermDefinition

The Box Problem

The box problem involves maximizing the volume of an open box constructed from a given rectangular sheet of material.

note

  • In the box problem, you are asked to maximize the volume of a box constructed from a rectangular sheet.

  • You know the dimensions of the sheet and the formula for volume of the box.

  • Relate the dimensions and volume by expressing each of the dimensions in terms of the length of a side of the square that must be removed from each of the corners. The resulting volume equation contains only one variable.

  • Set the derivative of the volume equal to 0 to find maximum point candidates. Notice that the values of s represent the lengths of the sides of the squares that must be removed to maximize the volume.

  • The quadratic formula gives two possible maximum
    candidates.

  • The greater of the two maximum candidates does not make sense. Removing this length from the dimensions of the box would result in negative length. Therefore, you only have one maximum candidate.

  • Make sure the candidate is a maximum by checking the sign of the derivative on either side of the point.

  • Notice that to the left of the point the derivative is positive. The function is increasing.

  • To the right the derivative is negative, so the function is
    decreasing.

  • If you increase to the left and decrease to the right, then the point in between must be a maximum.

  • Make sure that you answer the question! In this problem you were asked to find the dimensions of the square that must be removed from the corners. The square is 2.378 by 2.378. If you were asked to find the dimensions of the box the answer would be different.

A man is constructing a box out of a 20 × 16 sheet of metal. He opts to cut a 3 × 3 square off of each corner of the sheet and then fold the sides upwards. What is the area of the wasted material? Dimensions are in inches.

36 square inches

A particular company manufactures cardboard boxes of many different volumes. They always start with a flat sheet of cardboard. By cutting an equal sized square from each corner and folding the sides upwards, they make a box. If the cardboard sheet is 18 × 16 inches and you cut a 2 × 2 square from each corner, what would be the volume of the resulting box?

336 cubic inches

What is the greatest amount of sand you could put in a sandbox made from a 4 foot by 6 foot piece of plywood? Remember, the sandbox will be made by cutting equal sized squares from the corners and then “folding” the sides upwards.

8.45 cubic feet

What is the maximum volume of an enclosed rectangular box with surface area equal to 54 sq. ft and a height of 3 ft?

V = 27 cubic feet

A man is constructing an open-faced box from a rectangular sheet of metal. To do so, he cuts 4 equal sized squares from the corners of the sheet and then folds the remaining metal upwards to create the sides of a box. What is the maximum possible volume that he could hold with an 18 × 14 inch sheet of metal?

292.86 cubic inches

Little Louie is making a diorama for his 2nd grade teacher and you are going to help him. Unfortunately, you have no shoeboxes. In fact, the only material you have for the box itself is a pizza box left over from last week’s midnight munchie run. After cutting away the parts of the box that are too greasy to use, you are left with a sheet of cardboard with dimensions 14 × 8 in inches. How big must the squares you cut out of the sheet of cardboard be in order to maximize Little Louie’s diorama?

1.639 by 1.639 inches