An open box is made from a square piece of cardboard 48 inches on a side by cutting identical squares from the corners and turning up the sides Use this information to complete parts a through c Find and interpret $$ V(9) $$ 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 $$ V(9) $$ is $$ \square $$ cubic inches, when the side of the square cut from the center of each edge of the piece of cardboard is 9 inches B. The volume of the box $$ V(9) $$ is $$ \square $$ cubic inches, when the side of the square cut from each corner of the piece of cardboard is 9 inches. C. The volume of the box $$ V(9) $$ is $$ \square $$ cubic inches, when the side of the square cut from the center of the piece of cardboard is 9 inches D. The volume of the box $$ V(9) $$ is $$ \square $$ cubic inches, when the side of the square cut from each corner and the center of each edge of the piece of cardboard is 9 inches.

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UpStudy AI · Accepted Answer

To find the volume of the open box formed by cutting squares from the corners of a square piece of cardboard, we can follow these steps:

1. **Identify the dimensions of the cardboard**: The cardboard is a square with a side length of 48 inches.

2. **Define the variable**: Let $$ x $$ be the side length of the square cut from each corner of the cardboard.

3. **Determine the dimensions of the box**:
   - After cutting out squares of side length $$ x $$ from each corner, the new dimensions of the box will be:
     - Length: $$ 48 - 2x $$
     - Width: $$ 48 - 2x $$
     - Height: $$ x $$

4. **Write the volume function**: The volume $$ V $$ of the box can be expressed as:
   $$
   V(x) = (48 - 2x)(48 - 2x)(x) = (48 - 2x)^2 \cdot x
   $$

5. **Calculate $$ V(9) $$**: Now we will substitute $$ x = 9 $$ into the volume function:
   $$
   V(9) = (48 - 2 \cdot 9)^2 \cdot 9
   $$

Let's calculate this step by step:

- First, calculate $$ 48 - 2 \cdot 9 $$:
     $$
     48 - 18 = 30
     $$

- Now, square this result:
     $$
     30^2 = 900
     $$

- Finally, multiply by $$ 9 $$:
     $$
     V(9) = 900 \cdot 9 = 8100
     $$

Thus, the volume of the box when the side of the square cut from each corner is 9 inches is $$ 8100 $$ cubic inches.

Now, let's interpret the result:

The correct choice is:
**B. The volume of the box $$ V(9) $$ is $$ 8100 $$ cubic inches, when the side of the square cut from each corner of the piece of cardboard is 9 inches.**
The volume of the box when each corner is cut by a 9-inch square is 8100 cubic inches.

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