Prompt
you are given a rectangular sheet of cardboard that measures 11 in. by 8.5 in. (see the diagram below). a small square of the same size is cut from each corner, and each side folded up along the cuts to from a box with no lid.
1. anya thinks the cut should be 1.5 inches to create the greatest volume, while terrence thinks it should be 3 inches.
explain how both students can determine the formula for the volume of the box.
determine which student's suggestion would create the larger volume.
explain how there can be two different volumes when each student starts with the same size cardboard.
2. why is the value of x limited to 0 in. < x < 4.25 in.?
(quick response )

1. Anya's suggestion will create the larger volume

2. x is limited to 0 < x < 4.25 because 8.5 - 2x > 0

Step-by-step explanation:

- Lets explain how to solve the problem

- The dimensions of the cardboard are 11 inches by 8.5 inches

- A small square of the same size is cut from each corner, and each

 side folded up along the cuts to from a box with no lid

- The formed solid will be a rectangular solid of three dimensions,

  length , width and height

- The volume of the solid is V = length × width × height

1.

- Anya thinks the cut should be 1.5 inches to create the greatest

 volume, while Terrence thinks it should be 3 inches

- Lets find the formula for the volume of both

∵ The length of the sheet is 11 inches

∵ The width of the sheet is 8.5

- We will cut x from each corner

∴ The length of the base of the box = 11 - 2x

∴ The width of the base of the box = 8.5 - 2x

∵ The height of the box is x

∴ The volume of the box = (11 - 2x) × (8.5 - 2x) × x

∴ The formula for the volume of both is V = x(11 - 2x)(8.5 - 2x)

∵ Anya thinks the cut should be 1.5 inches

∴ x = 1.5

∴ V = 1.5(11 - 2×1.5)(8.5 - 2×1.5)

∴ V = 1.5(11 - 3)(8.5 - 3)

∴ V = 1.5(8)(5.5) = 66 inches³

* The volume of Anya's box is 66 inches³

∵ Terrence thinks the cut should be 3 inches

∴ x = 3

∴ V = 3(11 - 2×3)(8.5 - 2×3)

∴ V = 3(11 - 6)(8.5 - 6)

∴ V = 3(5)(2.5) = 37.5 inches³

∴ The volume of Terrence box is 37.5 inches³

* Anya's suggestion will create the larger volume

∵ The volume of the box depends on the dimensions of its base

   and its height

∴ We can make from the same cardboard many boxes with different

   dimensions by cutting small squares of the same size from its four

   corners with different sides (chose different values of x)

2.

∵ The small dimension of the cardboard is 8.5 inches

∵ We will cut x from its corners

∵ The new dimension will be 8.5 - 2x

- The new dimension must be greater than 0

∴ 8.5 - 2x > 0 ⇒ add 2x for both sides

∴ 8.5 > 2x ⇒ divide both sides by 2

∴ 4.25 > x

∴ x must be greater than 0 inch and smaller than 4.25 inches

* x is limited to 0 < x < 4.25 because 8.5 - 2x > 0

2

step-by-step explanation:

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