We are tasked with determining the size of the squares to be cut from the corners of a rectangular piece of cardboard, so that the resulting open-topped box has a volume of 1040 cm³.

Step-by-step approach:

1. Dimensions of the cardboard:

   • The rectangular piece of cardboard has dimensions of 24 cm by 30 cm.
   • Let $x$ be the side length of the squares cut from each corner.

2. Dimensions of the box:

   • After cutting out the squares, the dimensions of the base of the box will be reduced by $2x$ in both directions:
     • The length of the box will be $30 - 2x$ cm.
     • The width of the box will be $24 - 2x$ cm.
   • The height of the box will be $x$, since the squares are folded up to form the sides.

3. Volume equation: The volume $V$ of the box is given by the formula for the volume of a rectangular prism: $V = \text{length} \times \text{width} \times \text{height}$ Plugging in the expressions for the dimensions of the box: $1040 = (30 - 2x)(24 - 2x)(x)$ This simplifies to: $1040 = (720 - 108x + 4x^2)(x)$ Expanding further: $1040 = 720x - 108x^2 + 4x^3$ Rearranging this into a standard cubic equation: $4x^3 - 108x^2 + 720x - 1040 = 0$

4. Solving the cubic equation: We now solve the equation $4x^3 - 108x^2 + 720x - 1040 = 0$. Dividing the entire equation by 4 simplifies it to: $x^3 - 27x^2 + 180x - 260 = 0$ This cubic equation can be solved using numerical methods or by factoring/trial and error.

Approximate solution:

Using numerical methods (e.g., solving via graphing or a calculator), we find that $x \approx 5$.

Thus, the squares to be cut should have a side length of approximately 5 cm.

Would you like more details on the numerical solution or have any questions?

Related questions:

1. How do we solve cubic equations using factoring?
2. What if the volume of the box were different? How would that affect the size of the squares?
3. Can we solve the cubic equation using the Rational Root Theorem?
4. What happens to the box’s dimensions if we cut smaller squares?
5. How can the concept of optimization be applied here for maximum volume?

Tip:

In problems involving geometry, always carefully set up your equations with all known values, and ensure units are consistent throughout your calculations!