a box with an open top is to be constructed from a rectangular piece of cardboard with dimension 6 in. by 10 in. by cutting out equal squares of side x at each corner and then folding up the sides as shown in the figure. express the volume v of the box as a function of x

To find the volume \( V \) of the box as a function of \( x \), let's follow the steps described:

1. **Start with the original dimensions of the cardboard:**
   - Length = 10 inches
   - Width = 6 inches

2. **Cut out equal squares of side \( x \) from each corner:**
   - The size of each square is \( x \times x \).

3. **Modify the dimensions of the cardboard after cutting:**
   - The new length of the box after folding will be \( 10 - 2x \) (since squares are cut from both ends of the 10-inch side).
   - The new width of the box will be \( 6 - 2x \) (since squares are cut from both ends of the 6-inch side).

4. **Height of the box:**
   - The height of the box is equal to the side of the square cut out, which is \( x \).

5. **Volume \( V \) of the box:**
   - Volume is calculated by multiplying the length, width, and height of the box:
   \[
   V(x) = (10 - 2x) \cdot (6 - 2x) \cdot x
   \]

Now, let's expand this expression:

\[
V(x) = (10 - 2x)(6 - 2x)x
\]

Expanding the terms inside the parentheses first:

\[
(10 - 2x)(6 - 2x) = 60 - 20x - 12x + 4x^2 = 60 - 32x + 4x^2
\]

Then, multiply by \( x \):

\[
V(x) = x(60 - 32x + 4x^2)
\]

Simplify:

\[
V(x) = 4x^3 - 32x^2 + 60x
\]

So, the volume \( V \) of the box as a function of \( x \) is:

\[
\boxed{V(x) = 4x^3 - 32x^2 + 60x}
\]

### Do you want more details or have any questions?

Here are some related questions:

1. What is the maximum possible value of \( x \) for this box to exist?
2. How can we find the value of \( x \) that maximizes the volume of the box?
3. What would the volume be if \( x = 1 \)?
4. How does changing \( x \) affect the box's volume?
5. What are the dimensions of the box if \( x = 2 \)?

**Tip:** To maximize or minimize functions, consider taking the derivative and finding the critical points!

Learn how to calculate the volume of a box formed from a rectangular piece of cardboard by cutting out equal squares from each corner and folding up the sides. This problem involves applying geometric and algebraic principles to find the volume as a function of the side length of the squares cut out. Suitable for high school students.

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