To find the absolute minimum of the surface area function $S(x) = x^2 + \frac{360}{x}$, we need to first find its derivative.

Step 1: Find the derivative of $S(x)$

The function is:

$S(x) = x^2 + \frac{360}{x}$

To differentiate this with respect to $x$, we will apply basic rules of differentiation.

1. The derivative of $x^2$ is $2x$.
2. The derivative of $\frac{360}{x}$, which is $360x^{-1}$, is $-360x^{-2}$ or $-\frac{360}{x^2}$.

Thus, the derivative of $S(x)$ is:

$S'(x) = 2x - \frac{360}{x^2}$

Part 2: Find the critical points

To find the critical points, set the derivative equal to zero and solve for $x$:

$2x - \frac{360}{x^2} = 0$

Multiply both sides by $x^2$ to eliminate the denominator:

$2x^3 - 360 = 0$

Now solve for $x$:

$2x^3 = 360$ $x^3 = 180$ $x = \sqrt[3]{180}$

You can leave the answer as $x = \sqrt[3]{180}$, or you can approximate this value. Let's calculate it:

$x \approx 5.646$

This is the value of $x$ where the surface area function has a critical point.

Would you like to proceed with finding the minimum surface area and dimensions of the box, or do you need further clarification?

Here are five related questions for you to explore:

1. What is the second derivative test and how do we use it to confirm the minimum?
2. How can we determine the dimensions of the box for this minimum surface area?
3. How does the volume constraint (90 cubic feet) affect the surface area?
4. What are some real-world applications of optimizing surface areas for boxes with open tops?
5. Can we derive the volume formula for a box with a square base and open top from basic principles?

Tip: When working with optimization problems, it helps to graph the function to visually confirm that the critical point you found corresponds to a minimum or maximum.