To find the minimum cost for constructing the rectangular box, we need to consider the dimensions and the costs associated with each surface of the box.

Let:

• $x$ be the length,
• $y$ be the width, and
• $h$ be the height of the box.

Given:

1. Volume constraint: $x \cdot y \cdot h = 13$ cubic feet.
2. Cost per square foot:
   • Top: $9$ dollars per square foot
   • Bottom: $11$ dollars per square foot
   • Four sides: $4$ dollars per square foot each

Step 1: Calculate Surface Areas

• Top area: $x \cdot y$
• Bottom area: $x \cdot y$
• Four sides area: $2(x \cdot h) + 2(y \cdot h)$

Step 2: Calculate Cost for Each Part

1. Cost of the top: $9 \cdot (x \cdot y)$
2. Cost of the bottom: $11 \cdot (x \cdot y)$
3. Cost of the four sides: $4 \cdot [2(x \cdot h) + 2(y \cdot h)] = 8(x \cdot h + y \cdot h)$

Step 3: Total Cost Function

The total cost $C$ is given by: $C = 9(x \cdot y) + 11(x \cdot y) + 8(x \cdot h + y \cdot h)$ $C = 20(x \cdot y) + 8(x \cdot h + y \cdot h)$

Step 4: Substitute $h$ in Terms of $x$ and $y$

Using the volume constraint $x \cdot y \cdot h = 13$: $h = \frac{13}{x \cdot y}$

Substitute $h$ into the cost function: $C = 20(x \cdot y) + 8\left(x \cdot \frac{13}{x \cdot y} + y \cdot \frac{13}{x \cdot y}\right)$ $C = 20(x \cdot y) + 8 \cdot \frac{13(x + y)}{x \cdot y}$ $C = 20(x \cdot y) + \frac{104(x + y)}{x \cdot y}$

Step 5: Optimization

To find the minimum cost, we would take partial derivatives of $C$ with respect to $x$ and $y$, set them to zero, and solve for $x$ and $y$. This is a calculus-based optimization problem.

Let me proceed with these calculations to determine the minimum cost.The critical point that is real and relevant to our problem (since dimensions must be positive real values) is:

$x = y = \frac{26^{1/3} \cdot 5^{2/3}}{5}$

Let's calculate this approximate value and use it to determine the corresponding height $h$ and the minimum cost.The optimal dimensions for minimizing the cost of the box are approximately:

• $x = 1.73$ feet
• $y = 1.73$ feet
• $h = 4.33$ feet

The minimum cost to construct this box is approximately $180.09.

Would you like more details on how these values were derived, or do you have any further questions?

Here are five related questions to deepen your understanding:

1. How does the cost function change if the price for one side varies?
2. What if the volume constraint was different—how would that affect the minimum cost?
3. How can the method of Lagrange multipliers be applied to similar optimization problems?
4. Can we use a similar approach to minimize surface area for a fixed cost?
5. How do changes in unit costs impact the dimensions of the box?

Tip: For optimization problems with constraints, expressing one variable in terms of others using the constraint can simplify the solution process significantly.