Question

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.?

Answer 1

Cuboid

Anya's box will have larger volume as compared to Terence's box

x cannot be 0 as length of square cannot be 0.

x cannot be 4.25 and has to be less than 4.25 so that there should be enough breadth of cardboard to make the box (Height of box cannot be 0)

Step-by-step explanation:

Length l = 11 in and breadth b= 8.5 in

Square of equal sizes cut from 4 corners.Let the side of square is x

so l = x + x + y where y is remaining length, b = x + x + z where z is remaining breadth.

The resultant box after cutting the corner squares will have length as y and breadth as z and height will x

In case of Anya, x = 1.5 in

$l = 1.5 + 1.5 + y\\11 = 3 + y\\y = 11 - 3 = 8$

$b = 1.5 + 1.5 + z\\8.5 = 3 + z\\z = 8.5 - 3\\z = 5.5$

so volume for Anya = V

⇒ $V_{a}$ = 1.5 × 8 × 5.5 = 66 cubic inches

In case of Terence, x = 3 in

$l = 3 + 3 + y\\11 = 6 + y\\y = 11 - 6 = 5$

$b = 3 + 3 + z\\8.5 = 6 + z\\z = 8.5 - 6\\z = 2.5$

so volume for Terence= $V_{t}$

⇒ $V_{t}$ = 3 × 5 × 2.5 = 37.5 cubic inches

Hence, Anya's box will have larger volume as compared to Terence's box

x cannot be 0 as length of square cannot be 0.

x cannot be 4.25 and has to be less than 4.25 so that there should be enough breadth of cardboard to make the box (Height of box cannot be 0)

So X is limited to lie between 0 and 4.25 in