Question

A box is formed by cutting squares from the four corners of a 7"-wide by 9"-long sheet of paper and folding up the sides.

Let x represent the length of the side of the square cutout (in inches), and let V represent the volume of the box (in cubic inches).
Write a formula that expresses V in terms of x.
V=
Suppose the function f determines the volume of the box (in cubic inches) given a cutout length (in inches). Write a function formula for f.
f(x)=

Answer 1

The volume V of the box formed by cutting out squares of side x from a 7" by 9" sheet and folding the sides is given by V = (7 - 2x) × (9 - 2x) × x. The function formula for f(x) is thus f(x) = (7 - 2x) × (9 - 2x) × x.

Step-by-step explanation:

To find the volume V of the box after cutting out squares of side x from each corner of a 7" by 9" sheet of paper, we consider the new dimensions. The length and width will each be reduced by 2x because squares are cut out from both sides. Therefore, the new dimensions of the box, after folding, will be (7 - 2x) inches by (9 - 2x) inches, with a height of x inches.

The volume V of a box is given by its length times its width times its height. Thus, once the squares are cut out and the sides are folded up, the volume formula is:

V = (length - 2x) × (width - 2x) × (height)

V = (7 - 2x) × (9 - 2x) × x

Therefore, the function f(x) that determines the volume of the box given a cutout length x is:

f(x) = (7 - 2x) × (9 - 2x) × x