A 13 ft. by 4 ft. piece of plywood is being used to build an open-top box The box is formed by making equal sized square cutouts from two corners of the plywood at the ends of one of the 13-foot sides. After these squares are discarded, three more cuts are made and these pieces are "folded up" and secured to create three of the four vertical sides. When the open side is placed against a wall, the open top box is formed. What is an expression that defines a function h which determines the volume of the box in terms of the length x of the side of the square cutout?

To find an expression for the volume \( V \) of the box in terms of the side length \( x \) of the square cutout, let's go through the problem step by step.

### Step 1: Understand the Box Dimensions
- The plywood is originally \( 13 \) feet by \( 4 \) feet.
- Square cutouts with side length \( x \) are made from two corners on one of the 13-foot sides.

After cutting out the squares, the dimensions of the base of the box will be:
- **Length of the base:** \( 13 - 2x \) (because two squares, each with side \( x \), are cut out from the length)
- **Width of the base:** \( 4 \) feet (this dimension remains unchanged as the cuts are made along the length)
- **Height of the box:** \( x \) (this is the same as the side length of the square cutouts)

### Step 2: Volume Formula
The volume \( V \) of the box can be expressed as:
\[
V = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the dimensions we have:
\[
V(x) = (13 - 2x) \times 4 \times x
\]

### Step 3: Simplify the Expression
Now, simplify the expression for the volume:
\[
V(x) = 4x(13 - 2x)
\]
\[
V(x) = 52x - 8x^2
\]

### Final Answer
Thus, the function \( h(x) \) that determines the volume of the box in terms of the side length \( x \) of the square cutout is:
\[
h(x) = 52x - 8x^2
\]

This function \( h(x) \) gives the volume of the open-top box for any valid cutout side length \( x \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the maximum possible volume of the box?
2. What are the valid values of \( x \) for which the volume function \( h(x) \) makes sense?
3. How does the volume change as \( x \) increases?
4. Can you find the value of \( x \) that maximizes the volume of the box?
5. How would the function change if the plywood's dimensions were different?

**Tip:** To find the maximum volume, you can take the derivative of \( h(x) \) and set it to zero to solve for \( x \).

Learn how to calculate the volume of an open-top box made from plywood with detailed step-by-step solution. This problem involves geometric calculations and algebraic expressions, focusing on the volume formula for a rectangular prism. Suitable for Grades 9-12.

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