Q: If the original cardboard is $$19$$ inches by $$16$$ inches, and you want the box to have a height of $$3$$ inches, what will be the area of the base of the box?

$$(19 - 2 \times 3)(16 - 2 \times 3) = 13 \times 10 = 130 in$^2$$

Question 1

A rectangular piece of cardboard measures 19 inches by 16 inches. Squares of side $$x$$ inches are cut from each corner, and the sides are folded up to form an open box (as shown in the diagram). Which of the following expressions gives the volume $$V of $$the box as a function of $$x$$?

Accepted Answer

$$V(x) = x(19 - 2x)(16 - 2x)$$

Question 2

If the volume of the open box formed from the cardboard is required to be $$352$$ cubic inches, which equation must be solved to find the value of $$x$$?

Accepted Answer

$$x(19 - 2x)(16 - 2x) = 352$$

Question 3

Suppose you cut squares of side $$x = 4$$ inches from each corner of the cardboard. What will be the dimensions (length, width, and height) of the resulting box?

Accepted Answer

Length: $$11 in$$, Width: $$8 in$$, Height: $$4 in$$

Question 4

What is the maximum possible value for $$x ($$the side of the square cut from each corner) so that the box can still be formed from the $$19 in by$$ $$16 in $$cardboard?

Accepted Answer

$$x \u003c 8 in$$

Question 5

If $$x = 4$$ inches, what is the volume of the box formed from the $$19 in by$$ $$16 in $$cardboard?

Accepted Answer

$$352$$ $$in^3$

Question 6

Which of the following best describes the process shown in the diagram for forming the box?

Accepted Answer

Cutting equal squares from each corner and folding up the sides

Question 7

If the original cardboard is $$19$$ inches by $$16$$ inches, and you want the box to have a height of $$3$$ inches, what will be the area of the base of the box?

Accepted Answer

$$(19 - 2 	imes 3)(16 - 2 	imes 3) = 13 	imes 10 = 130 in$^2$$

Question 8

If the volume of the box is given by $$V(x) = x(19 - 2x)(16 - 2x)$$, what type of function is $$V(x)$$?

Accepted Answer

A cubic function

Question 9

If you want to maximize the volume of the box, which mathematical technique would you use?

Accepted Answer

Find the critical points of $$V(x) by $$setting its derivative to zero

Question 10

If the box must have a volume of $$352 in$^3$$, which of the following is a valid value for $$x$$?

Accepted Answer

$$x = 4 in$$

Maximizing Volume of an Open-Top Box Formed from a Rectangular Cardboard

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