A box is to be formed by cutting square pieces out of the corner of a rectangular piece of a 4" by 6" note card as shown below. The sides will then be folded up to make the box.

Part I: (4 points)
(A) Express the new length of the long side of the note card, once the two corners are removed.
(B) Express the new width of the short side of the note card, once the two corners are removed.

Part II:
Write function A(x) that defines the area of the bottom of the box once the corners are removed and the sides are folded up. (4 points)

Part III: (8 points)
(A) Suppose you want the bottom of your box to cover a total area of 16 in^2. Set up an equation in standard form that will help you find the size (x) of the corner you need to cut in order for your box to have this area.
(B) solve this equation and take note of any extraneous solutions. Explain why the answer is extraneous and clearly state the correct answer.

(A) To find the new length of the long side of the note card after cutting the corners, we subtract twice the length of the corner square from the original length.

New length of the long side = 6" - 2x, where x is the side length of the square cut from the corner.

(B) To find the new width of the short side of the note card after cutting the corners, we subtract twice the width of the corner square from the original width.

New width of the short side = 4" - 2x

Part II:

The function A(x) that defines the area of the bottom of the box after cutting the corners and folding up the sides can be calculated by multiplying the dimensions of the base.

A(x) = (6 - 2x)(4 - 2x)

Part III:

(A) Setting up an equation for the area of the box to be 16 in^2:

(6 - 2x)(4 - 2x) = 16

(B) Solving the equation:

Expanding and simplifying: 24 - 20x + 4x^2 = 16

Rearranging: 4x^2 - 20x + 8 = 0

Dividing by 4 to simplify: x^2 - 5x + 2 = 0

Using the quadratic formula, we find the solutions: x = (5 ± √17)/2

The extraneous solution is x = (5 - √17)/2 because it results in a negative value for x, which is not feasible in this context.

The correct answer is x = (5 + √17)/2.

Step-by-step explanation: