Let x, y, z be positive real numbers such that `x + y + z = 12` and `x^3y^4z^5 = (0.1)(600)^3`. Then `x^3+y^3+z^3` is

Options

A. 270
B. 258
C. 216
D. 342

Explanation

To solve the problem step by step, we need to find the value of \( x^3 + y^3 + z^3 \) given that \( x + y + z = 12 \) and \( x^3 y^4 z^5 = (0.1)(600)^3 \).

### Step 1: Set up the equations
We have two equations:
1. \( x + y + z = 12 \)
2. \( x^3 y^4 z^5 = 0.1 \times (600)^3 \)

### Step 2: Calculate \( 0.1 \times (600)^3 \)
First, we calculate \( 600^3 \):
\[ 600^3 = 216000000 \]
Now, multiplying by \( 0.1 \):
\[ 0.1 \times 600^3 = 0.1 \times 216000000 = 21600000 \]
Thus, we have:
\[ x^3 y^4 z^5 = 21600000 \]

### Step 3: Apply the Weighted AM-GM Inequality
We will use the Weighted Arithmetic Mean-Geometric Mean (AM-GM) inequality. The weights for \( x, y, z \) are \( 3, 4, 5 \) respectively. According to the AM-GM inequality:
\[ \frac{3 \cdot \frac{x}{3} + 4 \cdot \frac{y}{4} + 5 \cdot \frac{z}{5}}{3 + 4 + 5} \geq \left( \frac{x}{3} \right)^3 \left( \frac{y}{4} \right)^4 \left( \frac{z}{5} \right)^5 \]
This simplifies to:
\[ \frac{x + y + z}{12} \geq \left( \frac{x^3}{3^3} \cdot \frac{y^4}{4^4} \cdot \frac{z^5}{5^5} \right)^{\frac{1}{12}} \]

### Step 4: Substitute \( x + y + z \)
Substituting \( x + y + z = 12 \):
\[ 1 \geq \left( \frac{x^3 y^4 z^5}{3^3 \cdot 4^4 \cdot 5^5} \right)^{\frac{1}{12}} \]
Now substituting \( x^3 y^4 z^5 = 21600000 \):
\[ 1 \geq \left( \frac{21600000}{3^3 \cdot 4^4 \cdot 5^5} \right)^{\frac{1}{12}} \]

### Step 5: Calculate the denominator
Calculating \( 3^3 \cdot 4^4 \cdot 5^5 \):
\[ 3^3 = 27, \quad 4^4 = 256, \quad 5^5 = 3125 \]
Now, calculating the product:
\[ 3^3 \cdot 4^4 \cdot 5^5 = 27 \cdot 256 \cdot 3125 \]
Calculating step by step:
\[ 27 \cdot 256 = 6912 \]
\[ 6912 \cdot 3125 = 21600000 \]
Thus, we have:
\[ 3^3 \cdot 4^4 \cdot 5^5 = 21600000 \]

### Step 6: Apply AM-GM condition
Now substituting back:
\[ 1 \geq \left( \frac{21600000}{21600000} \right)^{\frac{1}{12}} = 1 \]
This equality holds, indicating that \( x, y, z \) must be proportional to their weights:
\[ \frac{x}{3} = \frac{y}{4} = \frac{z}{5} \]

### Step 7: Solve for \( x, y, z \)
Let \( k \) be the common ratio:
\[ x = 3k, \quad y = 4k, \quad z = 5k \]
Substituting into the sum:
\[ 3k + 4k + 5k = 12 \implies 12k = 12 \implies k = 1 \]
Thus:
\[ x = 3, \quad y = 4, \quad z = 5 \]

### Step 8: Calculate \( x^3 + y^3 + z^3 \)
Now we can calculate:
\[ x^3 + y^3 + z^3 = 3^3 + 4^3 + 5^3 \]
Calculating each term:
\[ 3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125 \]
Adding them together:
\[ 27 + 64 + 125 = 216 \]

### Final Answer
Thus, the value of \( x^3 + y^3 + z^3 \) is:
\[ \boxed{216} \]