An open box is made from a square piece of cardboard 48 inches on a side by cutting identical squares from the
comers and turning up the sides. Use this information to complete parts a through
Find and interpret Select the correct choice below and, if necessary, fill in the answer box to complete your
choice
(Simplify your answer. Type an integer or a decimal.)
A. The volume of the box is cubic inches, when the side of the square cut from the center of each edge
of the piece of cardboard is 8 inches
B. The volume of the box is cubic inches, when the side of the square cut from the center of the piece of
cardboard is 8 inches.
C. The volume of the box is cubic inches, when the side of the square cut from each comer of the piece
of cardboard is 8 inches
D. The volume of the box is cubic inches, when the side of the square cut from each comer and the
center of each edge of the piece of cardboard is 8 inches

Ask by Riley Logan. in the United States

Mar 12,2025

Question

An open box is made from a square piece of cardboard 48 inches on a side by cutting identical squares from the
comers and turning up the sides. Use this information to complete parts a through
Find and interpret Select the correct choice below and, if necessary, fill in the answer box to complete your
choice
(Simplify your answer. Type an integer or a decimal.)
A. The volume of the box is cubic inches, when the side of the square cut from the center of each edge
of the piece of cardboard is 8 inches
B. The volume of the box is cubic inches, when the side of the square cut from the center of the piece of
cardboard is 8 inches.
C. The volume of the box is cubic inches, when the side of the square cut from each comer of the piece
of cardboard is 8 inches
D. The volume of the box is cubic inches, when the side of the square cut from each comer and the
center of each edge of the piece of cardboard is 8 inches

Ask by Riley Logan. in the United States

Mar 12,2025

UpStudy AI · Accepted Answer

We start by letting $$ x $$ be the side length of the square cut from each corner. Since the original piece of cardboard is a 48‑inch square, cutting out squares of side $$ x $$ from each corner leaves a base with dimensions

$$
(48-2x) \times (48-2x).
$$

Since the sides are turned up to form the box, the height of the box is $$ x $$. Thus, the volume function $$ V(x) $$ is

$$
V(x)= x(48-2x)^2.
$$

To find $$ V(8) $$, we substitute $$ x=8 $$ into this expression:

$$
V(8)= 8\left(48-2(8)\right)^2= 8\left(48-16\right)^2= 8(32)^2.
$$

Since $$ 32^2=1024 $$, we have

$$
V(8)= 8\times 1024=8192.
$$

Therefore, $$ V(8)=8192 $$ cubic inches.

This means that the volume of the box is 8192 cubic inches when squares of side length 8 inches are cut from each corner of the cardboard. This corresponds to:

C. The volume of the box $$ V(8) $$ is 8192 cubic inches, when the side of the square cut from each corner of the piece of cardboard is 8 inches.
The volume of the box $$ V(8) $$ is 8,192 cubic inches when squares of side length 8 inches are cut from each corner of the cardboard.

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