Key Concepts and Formulas

• Arithmetic Mean - Geometric Mean (AM-GM) Inequality: For $n$ non-negative real numbers $a_1, a_2, \ldots, a_n$, $\frac{a_1 + a_2 + \ldots + a_n}{n} \ge \sqrt[n]{a_1 a_2 \ldots a_n}$
• Equality Condition for AM-GM: Equality holds if and only if $a_1 = a_2 = \ldots = a_n$. This condition is crucial for finding specific values of variables.

Step-by-Step Solution Step 1: Analyze the Problem and Identify the Strategy We are given a sum $x+y+z=12$ and a product $x^3 y^4 z^5 = (0.1)(600)^3$. We need to find $x^3+y^3+z^3$. The presence of a sum and a product, especially with powers, strongly suggests the application of the AM-GM inequality. The exponents in the product are 3, 4, and 5, and their sum is $3+4+5=12$, which matches the given sum $x+y+z=12$. This is a key indicator that we should apply AM-GM to a total of 12 terms.

Step 2: Construct the Terms for AM-GM To effectively use AM-GM and relate the given sum and product, we strategically choose terms such that their sum simplifies to $x+y+z$ and their product involves $x^3 y^4 z^5$. Based on the exponents (3, 4, 5), we select:

• 3 terms of $\frac{x}{3}$
• 4 terms of $\frac{y}{4}$
• 5 terms of $\frac{z}{5}$ The total number of terms is $n = 3+4+5=12$.

Step 3: Apply the AM-GM Inequality Applying the AM-GM inequality to these 12 terms: $\frac{\left(\frac{x}{3}\right) + \left(\frac{x}{3}\right) + \left(\frac{x}{3}\right) + \left(\frac{y}{4}\right) + \ldots + \left(\frac{y}{4}\right) + \left(\frac{z}{5}\right) + \ldots + \left(\frac{z}{5}\right)}{12} \ge \sqrt[12]{\left(\frac{x}{3}\right)^3 \left(\frac{y}{4}\right)^4 \left(\frac{z}{5}\right)^5}$ Simplifying the left side (the Arithmetic Mean): $\text{AM} = \frac{3 \cdot \frac{x}{3} + 4 \cdot \frac{y}{4} + 5 \cdot \frac{z}{5}}{12} = \frac{x+y+z}{12}$ Simplifying the right side (the Geometric Mean): $\text{GM} = \sqrt[12]{\frac{x^3}{3^3} \cdot \frac{y^4}{4^4} \cdot \frac{z^5}{5^5}} = \sqrt[12]{\frac{x^3 y^4 z^5}{3^3 \cdot 4^4 \cdot 5^5}}$ So, the inequality becomes: $\frac{x+y+z}{12} \ge \sqrt[12]{\frac{x^3 y^4 z^5}{3^3 \cdot 4^4 \cdot 5^5}}$

Step 4: Substitute Given Values and Evaluate We are given $x+y+z=12$. Let's evaluate the product term $x^3 y^4 z^5$: $x^3 y^4 z^5 = (0.1)(600)^3 = \frac{1}{10} \times (6 \times 10^2)^3 = \frac{1}{10} \times 6^3 \times (10^2)^3 = \frac{1}{10} \times 216 \times 10^6 = 216 \times 10^5$ Now, let's evaluate the denominator of the GM: $3^3 \cdot 4^4 \cdot 5^5 = 27 \cdot 256 \cdot 3125 = 6912 \cdot 3125 = 21,600,000 = 216 \times 10^5$ Substitute these values into the AM-GM inequality: $\frac{12}{12} \ge \sqrt[12]{\frac{216 \times 10^5}{216 \times 10^5}}$ $1 \ge \sqrt[12]{1}$ $1 \ge 1$ This result shows that the AM is exactly equal to the GM.

Step 5: Utilize the Equality Condition to Find x, y, z Since equality holds in the AM-GM inequality, all the terms we used must be equal: $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ Let this common ratio be $k$. Then: $x = 3k$ $y = 4k$ $z = 5k$ Substitute these into the given sum $x+y+z=12$: $3k + 4k + 5k = 12$ $12k = 12$ $k = 1$ Thus, the values of $x, y, z$ are: $x = 3(1) = 3$ $y = 4(1) = 4$ $z = 5(1) = 5$

Step 6: Calculate the Final Expression We need to find $x^3 + y^3 + z^3$: $x^3 + y^3 + z^3 = (3)^3 + (4)^3 + (5)^3$ $= 27 + 64 + 125$ $= 216$

Common Mistakes & Tips

• Incorrect Term Selection: The most common mistake is choosing the wrong terms for the AM-GM inequality. Always ensure the sum of terms simplifies to the given sum and the product of terms relates to the given product.
• Ignoring Equality Condition: If the problem asks for a specific value, the equality condition of AM-GM is almost always met. Failing to use it will prevent you from finding the unique values of variables.
• Calculation Errors: Be meticulous with calculations involving powers and large numbers, as a small error can lead to an incorrect final answer.

Summary This problem is a classic illustration of the AM-GM inequality. By strategically applying AM-GM to 12 terms constructed as $\frac{x}{3}$ (3 times), $\frac{y}{4}$ (4 times), and $\frac{z}{5}$ (5 times), we established a relationship between the given sum $x+y+z=12$ and the product $x^3 y^4 z^5$. The equality condition of the AM-GM inequality, which held true in this case, allowed us to determine the unique values of $x, y, z$ as 3, 4, and 5 respectively. Consequently, $x^3+y^3+z^3$ was calculated to be 216.

Final Answer The final answer is $\boxed{\text{216}}$ which corresponds to option (A).