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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.?

1.

Anya could find out her volume by multiplying 1.5*1.5*4 and then subracting that from 11*8.5.

Terrence could find out his volume by multiplying 3*3*4 and then subtracting that from 11*8.5

Anya's suggestion would maker a larger volume.

There can be 2 different volumes because someone might cut out more than the other person.

2.

I think there's something missing from this question.

1. Anya's suggestion

2. Length is always positive

Step-by-step explanation:

Volume = length × width × height

1.

Volumes:

Anya's approach

1.5 is the height of the box

Since 2 squares are cut, the dimensions of the base reduce be 2×1.5 = 3

Volume = (11 - 3) × (8.5 - 3) × 1.5

= 55 in³

Terrence's approach

3 is the height of the box

Since 2 squares are cut, the dimensions of the base reduce be 2×3 = 6

Volume = (11 - 6) × (8.5 - 6) × 3

= 37.5 in³

2.

Dimensions of the base are:

(8.5-2x) × (11 - 2x)

Since length cannot be negative:

x > 0

8.5 - 2x > 0

2x < 8.5

x < 4.25

11 - 2x > 0

x < 5.5

The set of values of x which satisfies all is:

0 < x < 4.25

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step-by-step explanation:

(x3 + 6x2 + 2x - 6) • (x - 1)

step-by-step explanation:

polynomial long division :

3.2     polynomial long division

dividing :   x4+5x3-4x2-8x+6

                              ("dividend")

by         :     x-1     ("divisor")

dividend     x4   +   5x3   -   4x2   -   8x   +   6

- divisor   * x3     x4   -   x3            

remainder         6x3   -   4x2   -   8x   +   6

- divisor   * 6x2         6x3   -   6x2        

remainder             2x2   -   8x   +   6

- divisor   * 2x1             2x2   -   2x    

remainder               -   6x   +   6

- divisor   * -6x0               -   6x   +   6

remainder                     0

quotient :   x3+6x2+2x-6   remainder:   0