Answer:
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
that it takes for the principal 
to double, so 
. Dividing both sides of the formula by 
and 
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