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The problem in the image deals with optimizing the volume of an open box, where squares of side length \( x \) are cut from the corners of a rectangular piece of card that is 40 cm by 10 cm, as shown in the diagrams (Figure 1 and Figure 2). We are asked to show that the volume \( V \) of the box is given by:

\[
V = 4x^3 - 100x^2 + 400x
\]

### Steps to derive the volume equation:

1. **Dimensions of the base of the box:**
   - After cutting squares with side length \( x \) from each corner, the dimensions of the base become:
     \[
     \text{Length} = 40 - 2x
     \]
     \[
     \text{Width} = 10 - 2x
     \]

2. **Height of the box:**
   - The height of the box is the same as the side length of the squares cut out, which is \( x \).

3. **Volume formula:**
   - The volume of a box is given by:
     \[
     V = \text{Length} \times \text{Width} \times \text{Height}
     \]
     Substituting the expressions for the length, width, and height:
     \[
     V = (40 - 2x)(10 - 2x)(x)
     \]

4. **Expanding the expression:**
   - First, expand \( (40 - 2x)(10 - 2x) \):
     \[
     (40 - 2x)(10 - 2x) = 400 - 80x - 20x + 4x^2 = 400 - 100x + 4x^2
     \]
   - Now multiply this by \( x \) (the height):
     \[
     V = x(400 - 100x + 4x^2) = 400x - 100x^2 + 4x^3
     \]

5. **Final expression:**
   \[
   V = 4x^3 - 100x^2 + 400x
   \]
This matches the given volume expression.

---

Would you like further details on the derivation, or do you have any other questions?

Here are 5 related questions to further your understanding:

1. How do you calculate the maximum volume for the box?
2. What constraints should \( x \) satisfy for the box to exist?
3. How does changing the dimensions of the rectangular sheet affect the volume formula?
4. Can you find the derivative of \( V(x) \) to find the critical points?
5. What is the practical interpretation of each term in the volume expression \( 4x^3 - 100x^2 + 400x \)?

**Tip:** When optimizing volumes or areas, always carefully identify the dimensions that change after cutting or folding parts of the shape.

Show that the volume of the open box is given by V = 4x^3 - 100x^2 + 400x, where squares of side length x cm are cut out from each corner of a rectangular piece of card measuring 40 cm by 10 cm.

Learn how to derive the volume formula V = 4x^3 - 100x^2 + 400x for an open box made by cutting squares from each corner of a rectangular piece of card. The rectangular sheet measures 40 cm by 10 cm, and this problem explores polynomial expansion and volume optimization concepts. Suitable for Grades 10-12.

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