To solve this problem, let's go step-by-step.

1. Given information:

   • The box has a square base with side length $x$.
   • The height of the box is $y$.
   • The box has an open top.
   • The volume of the box is given as $78732 \, \text{cm}^3$.

2. Objective:

   • Find the dimensions of the box that minimize the surface area (material used) of the box.
   • First, we need to express the surface area $A(x)$ in terms of $x$ only.

Step 1: Express the Volume Constraint

Since the box has a square base and height $y$, the volume $V$ is: $V = x^2 \cdot y$ Given that $V = 78732 \, \text{cm}^3$, we can write: $x^2 \cdot y = 78732$ Solving for $y$ in terms of $x$: $y = \frac{78732}{x^2}$

Step 2: Set Up the Surface Area Function

The surface area $A$ of the box, with an open top, includes:

• The area of the square base: $x^2$.
• The area of the four sides: $4 \cdot x \cdot y$.

So, the surface area $A(x)$ in terms of $x$ and $y$ is: $A(x) = x^2 + 4xy$ Substitute $y = \frac{78732}{x^2}$ into this equation: $A(x) = x^2 + 4x \cdot \frac{78732}{x^2}$ Simplify: $A(x) = x^2 + \frac{314928}{x}$

Final Answer for $A(x)$:

$A(x) = x^2 + \frac{314928}{x}$

This is the formula for the surface area in terms of $x$. To minimize the material used, we would take the derivative of $A(x)$ with respect to $x$ and set it to zero to find the critical points, then check which value of $x$ minimizes $A(x)$.

Would you like to continue with the optimization to find the exact dimensions that minimize the surface area?

Here are some additional questions to consider:

1. How do you find the minimum value of a function?
2. Why is it important to express $y$ in terms of $x$ using the volume constraint?
3. What would change in the formula if the box had a closed top?
4. How does the volume constraint influence the shape of the box?
5. Why might minimizing surface area be important in a practical setting?

Tip: Remember that minimizing surface area is key in reducing material costs for production, making this method valuable in real-world applications.