To solve this problem, we are given the following conditions:

1. We have a rectangular box with dimensions $x$, $y$, and $z$.
2. The volume of the box is $V = 3375$, which means: $x \cdot y \cdot z = 3375$
3. We need to minimize the surface area of the box. The surface area $A$ is given by: $A = 2(xy + yz + xz)$

Step-by-Step Solution

1. Volume Constraint: Solve the volume constraint for one variable. Let's solve for $z$: $z = \frac{3375}{xy}$

2. Substitute into the Surface Area Formula: $A = 2\left(xy + y \left(\frac{3375}{xy}\right) + x\left(\frac{3375}{xy}\right) \right)$ Simplify this to: $A = 2\left(xy + \frac{3375}{x} + \frac{3375}{y}\right)$

3. Optimization: To minimize the surface area, take the partial derivatives of $A$ with respect to $x$ and $y$, set them equal to zero, and solve the system of equations.

Solution

Since this problem involves symmetry in minimizing the surface area, an optimal solution often results in a box that is close to being a cube. For a cube: $x = y = z$

Using the volume equation: $x^3 = 3375$ Solve for $x$: $x = \sqrt[3]{3375} = 15$

Thus, the dimensions that minimize the surface area are: $x = 15, \quad y = 15, \quad z = 15$

Answer

The optimal dimensions are:

• $x = 15$
• $y = 15$
• $z = 15$

Do you want further details or have any other questions?

Here are 5 related questions that can expand the concept:

1. What if the box had a different volume constraint? How would the solution change?
2. How would the calculation vary if the box had an open top instead of being closed?
3. What is the impact of different ratios between $x$, $y$, and $z$ on the surface area for a fixed volume?
4. Can we use calculus to derive a general formula for minimizing the surface area of a box with any given volume?
5. How would this problem be approached if minimizing the perimeter instead of the surface area?

Tip: For optimization problems involving geometry, symmetry often provides an efficient solution, as shapes like cubes or spheres often lead to optimal results.