The problem is asking for the minimum sum of the cubes of two numbers whose sum is 13. Let the two numbers be $x$ and $y$, such that:

$x + y = 13$

We need to minimize $S = x^3 + y^3$. Using the constraint $x + y = 13$, we can express $y$ in terms of $x$:

$y = 13 - x$

Substitute this into $S$:

$S = x^3 + (13 - x)^3$

Simplify $(13 - x)^3$ using the binomial expansion:

$(13 - x)^3 = 13^3 - 3(13^2)x + 3(13)x^2 - x^3$

Substitute this back into $S$:

$S = x^3 + \big(13^3 - 3(13^2)x + 3(13)x^2 - x^3\big)$

Simplify:

$S = 13^3 - 3(13^2)x + 3(13)x^2$

Next, to minimize $S$, take the derivative with respect to $x$ and set it to zero:

$\frac{dS}{dx} = -3(13^2) + 6(13)x = 0$

Simplify:

$-3(169) + 78x = 0$

$78x = 507$

$x = \frac{507}{78} = 6.5$

Substitute $x = 6.5$ back into $y = 13 - x$:

$y = 13 - 6.5 = 6.5$

Thus, $x = y = 6.5$.

Finally, calculate the minimum sum of the cubes:

$S = x^3 + y^3 = 6.5^3 + 6.5^3 = 2(6.5^3)$

$6.5^3 = 6.5 \times 6.5 \times 6.5 = 274.625$

$S = 2 \times 274.625 = 549.25$

Final Answer:

The minimum sum of the cubes is:

$\boxed{549.25}$

Would you like me to expand or verify any steps?

Relative Questions:

1. Why does minimizing $x^3 + y^3$ require calculus?
2. How does the constraint $x + y = 13$ influence the optimization process?
3. Can you explain why $x = y$ results in the minimum value?
4. What is the geometric interpretation of this problem?
5. How would the solution change if the constraint was different?

Tip: Always check for symmetry in optimization problems—it often simplifies calculations.