We are given two positive numbers, say $x$ and $y$, such that:

$x \cdot y = 84$

We are tasked with finding the smallest possible value of:

$S = x^2 + y^3$

subject to the constraint $x \cdot y = 84$. To solve this, we can express $y$ in terms of $x$ using the constraint:

$y = \frac{84}{x}$

Substitute this into the expression for $S$:

$S(x) = x^2 + \left( \frac{84}{x} \right)^3$

Now we have:

$S(x) = x^2 + \frac{84^3}{x^3}$

Next, we minimize $S(x)$ by finding the critical points. First, take the derivative of $S(x)$ with respect to $x$, set it equal to zero, and solve for $x$.

Let's calculate this now.The real critical point for $x$ is $2 \cdot 27783^{1/5}$. We will now calculate the numerical value of this real solution and substitute it back into the expression for $S(x)$ to find the smallest sum $S$. Let's proceed with that calculation.The real critical point occurs when $x \approx 15.5$. Substituting this value into the expression for $S(x)$, we find that the smallest possible sum of the square of one number and the cube of the other is approximately:

$S \approx 399.4$

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions for further practice:

1. How would the result change if the product of the two numbers was different from 84?
2. What is the geometric interpretation of minimizing this sum in terms of curves?
3. Can this problem be solved using a different method, such as Lagrange multipliers?
4. What happens to the function $S(x)$ as $x \to 0$ or $x \to \infty$?
5. How would rounding affect the accuracy of this solution?

Tip: For optimization problems with a product constraint, rewriting one variable in terms of the other can simplify finding critical points.